Dorothée Boccanfuso

Céline De Quatrebarbes
email: celine.dequatrebarbes@gmail.com

Luc Savard

Gredi

Can the removal of VAT Exemptions support the Poor? The Case of Niger

In order to have the public funds necessary for its development, Niger is examining the possibility of expanding its Value-added tax (VAT) tax base to exempted goods and basic food products. This proposal has prompted violent opposition leading to the question of the social impacts of taxation. The first micro-macro computable general equilibrium model of Niger's actual economy has been developed. This model allows analysis of the social impact and distributional analysis of the following VAT structures: a pure VAT structure, a structure maintaining certain exemptions, and a multiple-rate VAT structure. The model's results shows that although restoring the VAT rate would be socially costly compared to the initial situation, the distributional impact of the VAT differs according to the system implemented in the country. Maintaining VAT exemptions in food crop agriculture sectors associated with a tax base expansion in the remaining sectors will increase public revenue while taking into account the national goal of poverty reduction. The net social impact of exoneration depends on the economic structure of the concerned sector. If the national goal is the end of exemption, the model shows that applying a pure VAT conforming to the theory is preferable in terms of economic growth whereas applying a reduced-rate on food crop agriculture lightens the social impact of the end of exemptions compared to a single rate. Production food crop agriculture -8.85 -4.85 -0.26 -2.72 -2.91 export agriculture -3.66 -2.56 -3.97 -3.25 -3.95 industry 20.67 11.66 10.08 11.06 13.62 services -7.03 -3.68 -4.93 -4.34 -5.73 trade 0.21 0.15 -0.42 -0.12 -0.24 Labour demand -1.95 0 -0.1 -0.11 -0.74 Wage rate food crop agriculture -9.43 -5.17 -0.28 -2.91 -3.10 export agriculture -4.01 -2.81 -4.35 -3.56 -4.34 industry 21.07 11.87 10.26 11.26 13.86 services -7.58 -3.98 -5.33 -4.69 -6.19 trade 0.22 0.15 -0.44 -0.13 -0.25 Remuneration of capital food crop agriculture -16.86 -8.47 -0.56 -4.90 -5.82 export agriculture -8.42 -4.64 -7.24 -5.97 -7.81 industry 34.85 20.56 17.56 19.33 23.25 services -14.02 -6.54 -8.82 -7.79 -10.77 trade -1.95 -8.47 -0.10 -0.11 -0.74 Foreign trade Export export agriculture -2.35 -2.06 -3.37 -2.68 -3.10 industry 23.04 13.06 11.49 12.48 15.35 Import food crop agriculture -11.28 -5.59 -0.71 -3.38 -4.06 export agriculture -7.47 -4.02 -5.73 -4.91 -6.44 industry 13.03 7.10 5.48 6.44 7.98 Domestic demand food crop agriculture -8.85 -4.85 -0.26 -2.72 -2.91 export agriculture -4.08 -2.72 -4.16 -3.43 -4.23 industry 19.61 11.04 9.45 10.43 12.84 services -7.03 -3.68 -4.93 -4.34 -5.73

Introduction

Niger has reinforced its value-added tax (VAT) system in view of successfully completing its fiscal transition and mobilizing resources required for its development. According to the available statistics [START_REF] Uemoa | Report : l'évaluation de l'application des directives de l'UEMOA, portant harmonisation des législations des Etats membres en matière de taxe sur la valeur ajoutée et de droits d'accises[END_REF], Niger nonetheless remains one of the WAEMU countries with the lowest overall public tax collection rates. 1 In 2009 Niger had a fiscal pressure rate of 13.9% and an average of 10.1% from 1998 to 2005 which is lower than the average of 14.08% of the GDP over the same period in the WAEMU. 2 In 2005, the VAT revenues were roughly 5% of the GDP, one of the lowest tax ratios in Africa. The VAT efficiency ratio constructed by [START_REF] Ebrill | The Modern VAT Washington[END_REF] 3 gives an average of 27% for Sub-Saharan African countries and 24% for Niger in 2005. It varies between 23% and 36% for the other WAEMU countries, except for Togo and Senegal whom have a much better performance.

Faced with this situation and in view of improving the returns of the internal indirect tax, the Nigerien government has tried to reduce VAT exemptions by extending the tax base to include basic food products. Exemption of these consumer goods indeed yields significant tax loss. Consumer goods represent an important share of transactions in the country but are often exempted or exchanged in the under-taxed traditional sector. Besides an increase in public revenue, the end of VAT exemptions should restore economic neutrality.

In a national and international poverty reduction strategies context it is important to assess the social impacts of this type of reform and to understand whether reinforcing public revenues might hamper efforts made to reduce poverty. The relation existing between the increase in public revenues and the establishment of effective poverty reduction programs is not always confirmed in the short term [START_REF] Filmer | Health policy in poor countries: weak links in the chain[END_REF][START_REF] Gauthier | Methodological Approaches to the Study of Institutions and Service Delivery: A Review of PETS, QSDS and CRCS, African Economic Research Consortium framework paper[END_REF]. This enables the hypothesis that the social objectives of the country should be taken into account right from the definition of the characteristics of public tax collection. The increase in fiscal pressure can be seen as legitimate by the population only if it is quickly associated with effective public expenditure allowing an improvement in the welfare of populations and a reduction in 1 It is important to remember, however, that current comparison in terms of level of tax collection between WAEMU countries is somewhat biased by the lack of harmonization of ways to calculate the level of fiscal pressure. The rate of fiscal pressure is generally calculated by the relation between total tax revenues and the GDP, but the difference is due to the composition of the numerator. The comparison can be solid only when all countries use the same calculation basis, which hardly seems to be the case today. 2 Niger, Finance ministry, Institut National de la Statistique, (2008). Report « L'analyse de la politique fiscale au Niger et ses impacts sur la pauvreté ». 3 [START_REF] Ebrill | The Modern VAT Washington[END_REF] have advanced efficiency differences of the VAT according to the economic and social structures of countries and have attempted to classify the countries and regions of the world according to VAT efficiency. The traditional measure is the "efficiency ratio"-the ratio of VAT revenues to the GDP divided by the standard VAT rate, that is, the single rate normally applied over all products. The efficiency ratio of a VAT system with satisfactory functioning is situated between 40% and 50% poverty. With this hypothesis, it is important to analyze the impact of the end of exemptions in an economy like Niger and define in what way a VAT structure could support the poor. This analysis primarily seeks to determine the household categories most affected in terms of increased poverty and inequalities. The distributional consequences of introducing various tax systems was evaluated by using a computable general equilibrium model (CGEM) representing the Nigerien economy linked to a micro-simulation model [START_REF] Decaluwe | Poverty Analysis Within a General Equilibrium Framework[END_REF].

The remainder of this paper is split up into four sections. In the first, we present the characteristics of the VAT structure in Niger and some assumptions about the social incidence of the end of VAT exemptions in Niger. The micro macro CGE model is presented in the second section. The third focuses on the definition of the VAT scenarios and the analysis of the aforesaid scenario's simulation. In the last section, we make some concluding comments.

The Nigerien choices regarding internal indirect taxation structure:

a. An imperfect VAT structure regardless to the theory:

The VAT was established in Niger on January 1, 1986. At the beginning, there were three VAT rates: 35%, 25%, and 15%. In 1994, these rates respectively became 24%, 17%, and 10%. In 1990, five years after the implementation of the VAT, comparable collection had decrease by more than 40%. This decline cannot be explained by movement in GDP which increased but by the importance increase of transaction exempt from VAT. Since 1998, Niger has had to conform to WAEMU directives regarding indirect taxation:4 choosing a single VAT rate ranging from 15% to 20%, harmonizing taxable goods and exemptions, possibly applying the VAT to the agricultural sector, and excluding transport activities. One of the objectives is to increase the taxation of the informal sector. Starting in January 2000, Niger thus implemented a single-rate VAT of 17%. In May 2000, with the establishment of the common external tariff and the expected decrease in customs revenues, the VAT rate climbed to 18% then to 19% in 2002.

The imperfections of the Nigerien VAT system are numerous compared to a "pure" VAT as prescribed by economic theories [START_REF] Chambas | TVA et transition fiscale en Afrique : les nouveaux enjeux[END_REF][START_REF] Ebrill | The Modern VAT Washington[END_REF]. Between 2003 and2005, legal exemptions still represented on average more than 43% of the internal VAT yield and exempted consumptions represented 48.6% of the consumption of Nigerien households living in Niamey. In rural areas, the VAT base represents only 43.1% of final consumption due to the VAT does not apply only to monetary consumptions. 5 It should be noted that VAT exemptions do not concern only commodities consumed by poor populations but increasingly extend to the industrial sector (mines and agro-food industry) for a more or less lengthy period. In addition to reducing the tax base, VAT exemptions create distortions within the economy that could have consequences on the economic level activity and on the poverty level.

b. The economic and social implications of exemptions

VAT exemptions and production

To avoid creating economic distortions, the VAT should theoretically be neutral in the production and distribution chain (for the producer, wholesaler, and retailer). Exemptions occur when output is untaxed but input tax is not recoverable. Exemptions break the VAT chain and create a cascading effect. The tax has to be paid by the producer using the exempted item, and no longer by the final consumer. This cascade effect increases production costs and creates distortions in the production choices. Contrary to common belief, the VAT6 no longer equals a consumption tax when exemptions are granted. The only way for the producer to remain neutral when faced with VAT exemptions would be to be able to charge this additional tax burden to the consumer by raising the sales price. This behaviour would be possible if the supply was very elastic and if the producer could fix the price.

Likewise, when the zero VAT rate on exported goods is not applied, the VAT credits of exporting firms are not reimbursed by the government and the problem is the same as in the presence of exemptions. The VAT cascading effect, which leads to the competitiveness of local producers in terms of international trading, creates a negative effective protection [START_REF] Bonjean | Le paradoxe de la fiscalité agricole en Afrique subsaharienne[END_REF]. In this case, the VAT has functioned largely as a tax on exports. The existence of informal sector can also create distortion in VAT application (Emran and Stigltiz, 2005). Indeed, the informal sector is taxed by the VAT when it purchases its inputs from the formal sector: it then cannot obtain any credit refund for its tax credits because it is non-taxable.

The reduction in exemptions and effective application of the zero VAT rate should enable an increase in the country's economic efficiency and hence improve allocations of resources.

From a social standpoint, any additional burden for the producer impacts the poverty level of the country by provoking a slow-down in the production system and a decrease in revenues from the production. Nevertheless, the end of exemptions can have an ambiguous effect in terms of poverty, as the latter also depends on households' ability to access basic goods.

VAT exemptions and consumption

A "pure" VAT must also be neutral for the consumer. A uniform taxation avoids distortion between prices of goods available on the territory. This is no longer the case when one of the goods is exempted or subject to a reduced rate. The relative decrease in retail price of this good compared with other goods subject to a full rate can make it preferable for other substitutable goods and thus modifies the economic structure of the demand.

From a social point of view, the VAT does not take into account each individual's ability to pay. In this context, exemptions can improve the country's situation by enabling consumers to have access to basic goods at lower prices. In a context of agricultural prices instability7 , an ad valorem tax like VAT can quickly increase tax burdens in household budgets.. In a country where 62.1% of the population lives below the poverty line (QUIBB_ 2005) and 34% in extreme poverty, where subsidies targeted at the level of household income are nearly inexistent, taxation can constitute an instrument to fight poverty. The progressivity of this type of measure is still limited compared to subsidies [START_REF] Alderman | Subventions en tant que filets sociaux de sécurité : efficacité et défis, Série d'introduction aux filets sociaux de sécurité[END_REF][START_REF] Ebrill | The Modern VAT Washington[END_REF]. Indeed, exemptions apply to a consumer good and not to a targeted population. The social effect thus depends on the consumption structure of the various social groups. The tax burden on the consumer is all the heavier as consumers demand is inelastic, which is the case with basic food products.

VAT exemptions thus have an ambiguous impact on the poverty level. The distribution of the tax burden depends on the structure of the market as well as the nature of the social groups concerned by the exemptions. The levels of production and consumption are interdependent, and the poverty situations of households are a function of their activity sector and their consumption structure.

VAT exemption and public revenue:

First, the problem of VAT application creates a gap between the VAT nominal rate and the effective VAT rate. Secondly, the tax expenditure brought about by VAT exemptions is difficult to evaluate [START_REF] Gautier | Les paradoxes de la TVA dans un pays en développement : l'illustration de l'industrie malgache[END_REF]: "Although the government loses revenues on the sales of non-taxable firms, it completely reclaims the non-exempted inputs of these same firms." The effect therefore depends on the net value of the tax base. 8 In addition, the various systems can significantly disrupt the application of legislation and lead to increased fraud.

The end of exemption forces the government to choose a new VAT structure supporting the poor: that is a uniform rate structure close to a pure VAT structure or the introduction of a reduce rate on large consumption goods. Lots of studies present the qualities of the both system [START_REF] Gautier | Taxation optimale et réformes fiscales dans les PED: une revue de littérature tropicalisée[END_REF][START_REF] Ebrill | The Modern VAT Washington[END_REF].

A computable general equilibrium model to study the social impact of expanding the tax base

Computable general equilibrium models (CGEM) have often been used to evaluate the macroeconomic impact of a tax reform, as presented in the literature review of [START_REF] Shoven | Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey[END_REF] or [START_REF] Burgess | Taxation and development[END_REF]. Lots of MEGC have been used to study the impact of trade tax decrease [START_REF] Clarete | Comparing the marginal welfare costs of commodity and trade taxes[END_REF][START_REF] Cockburn | Trade Liberalisation and Poverty in Nepal A Computable General Equilibrium Micro Simulation Analysis[END_REF] and more generally the impact of a change in national tax structure [START_REF] Toh | An evaluation of the 1994 tax reform in China using a general equilibrium model[END_REF]. VAT is usually conceptualized as a uniform consumption tax [START_REF] Serra-Puche | A general Equilibrium Model for Mexican Economy[END_REF]Ballard and[START_REF] Ballard | A general Equilibrium Model for Tax Policy Evaluation[END_REF][START_REF] Bovenberg | Indirect Taxation in Developing Countries: A General Equilibrium Approach[END_REF]. Little CGEM take into account the intern complexity of VAT system while literature talks about it [START_REF] Ebrill | The Modern VAT Washington[END_REF]. [START_REF] Bovenberg | Indirect Taxation in Developing Countries: A General Equilibrium Approach[END_REF], [START_REF] Emini | Analyse de l'incidence d'une TVA imparfaite à l'aide d'un modèle calculable d'équilibre général : Application au cas Camerounais[END_REF] Giesecke and Nhi (2010) have examined the impact of the various VAT systems in a given economy. They discuss the possibility of moving away from a uniform taxation rate by multiplying rates and the possibility to exempt goods. The key variables in the analysis are consumption, sectoral factor reallocation, production, government revenue, and household welfare. [START_REF] Emini | Analyse de l'incidence d'une TVA imparfaite à l'aide d'un modèle calculable d'équilibre général : Application au cas Camerounais[END_REF] showed that by postulating budget neutrality with a lower level of taxation than the one set down in the finance act, re-establishing VAT neutrality by expanding the tax base enables economic expansion and an increase in household welfare compared to the initial situation. An increase in inequalities between household categories must nonetheless be noted. When Emini rejects the hypothesis of budget neutrality and reestablishes the VAT at the official rate, he shows that the sudden increase in VAT leads to a strong loss in reallocation, especially for sectors with a weak ex ante tax burden. In this context, household welfare decreases, as inequalities.

The model used aims to take into account imperfections at the level of the VAT system by modelling, starting with the initial situation, the existence of VAT exemptions in certain sectors of the Nigerien economy. It also aims to compare the social impact of different VAT systems ranging from single-rate to multiple-rate and to the preservation of certain exemptions on agricultural goods. We aim to analyse the impact of these changes on poverty and inequalities while taking into account heterogeneity in the structure of household income and expenses. We have constructed a CGEM micro-simulation to achieve this goal. This is the first time this technique has been used to address this type of problems, as authors frequently use CGEMs with representative agents9 .

Nigerien data

The model used data from a social accounting matrix (SAM) of the Niger economy for 2004 and a household survey Questionnaire unifié des Indicateurs de base du bien-être (QUIBB) for 2005. The information on Nigerien tax structure is provided by the country's fiscal office for the year 2006.

The SAM has been aggregated to cover five sectors: food crop agriculture, export agriculture, industry10 and services, the retail branch and transportation, as well as the production of public services by the government. The aggregation of the SAM provides a way to make up for the weakness in tax information. The distinction between food crop agriculture and export agriculture offers a way to take into account specific characteristics of these two agricultural sectors a primarily informal part of agriculture, an under-taxed locally consumed, and a primarily exported form of agriculture.

Primary sector production is predominant in Niger. Agricultural production, including export agriculture and food crop agriculture, represents 40.6% of national value added. Services constitutes 23.9% of the country's total value added, followed by industry with 19.3%.

Extraction industries with the mining sector, gold and uranium, produce 9% of total value added. Agriculture has a high value added rate, respectively 91.7% for non-exporting food crop agriculture and 85.4% for export agriculture, based on the weak use of intermediate inputs. Trade with the rest of the world is concentrated in the industry sector (68.6% of exports and 85.1% of imports). Export agriculture represents 31.4% of exports and food crop agriculture, 10.4% of imports. The penetration rate of imports in the country is 18.53% and these imports are foremost made up of industrial goods, but available goods on the territory are nevertheless mostly locally produced. The labour factor receives the largest share of value added, primarily from the agricultural sectors. The informal sector (principally agriculture and trade) constitutes 70 to 80% of traded GDP, employs a large share of the population, and contributes to social stability by enabling populations to have a source of income. Households acquire their income from labour (91.3%) and capital (5.3%) but also transfers from government (2.5%) and the rest of the world (1%). Their main expenditure item is food.

The SAM has been disaggregated in order to distinguish VAT collected by the customs and by the general direction of taxes.11 According to the information from the SAM, it can be seen that the level of VAT collected is roughly 22% of the sum of payable VAT when considering a pure VAT at a 19% rate touching on only final consumption, representing a little less than 5% of GDP and 37% of total tax revenues, which is consistent with the available statistics. The VAT levied on customs represents 60% of total VAT revenues. The average VAT rate is thus significantly below the legal VAT rate, the difference being due to two factors: the high sum of VAT exemptions and tax evasion. Most internal VAT revenues come from the taxation of industrial goods. As for taxes on foreign trade, the sectors most affected are chemical, oil, and manufactured products. Direct taxes do not concern the entire population and the tax burden especially concerns the workers in the formal private and public sectors. Individual entrepreneurs in the informal sector also pay a direct tax (for example the overall trading tax), although the sum of these tax revenues is weak. (Data on the structure of the Nigerien economy based on the SAM can be found in Appendix 1.)

Characteristics of the computable general equilibrium model

The model draws on EXTER by Decaluwé, et al (2001) as a starting point. It has been modified to take into account the specificities of the Nigerien economy and takes up the SAM structure presented above. Each sector produces a single good. As a result, there is the same number of goods produced as there are production sectors, with the addition of oil, which is consumed but not locally produced. Only the "export agriculture" and "industry" sectors take part in international trade. The "trade," "service," and "food crop agriculture" sectors are non-tradable.

The production technology is represented by constant return to scale which combine labor and capital. We suppose full employment of factor. Labor is mobile from one sector to another while capital is fixed. The fixity of capital can be explained in Niger by the weak development of financial markets and the difficulty of transformation of capital in the short-to medium-term. Hence, the model considers the short-to medium term influences of a tax policy. The level of intermediate consumption is linked to production by fixed technical coefficients. The functional form describing the level of value added is a production function with constant substitution elasticity taking into account labour/capital substitution. Producers in each sector maximize profits subject to their technology constraint and determine optimal factor's demand.

The double Armington assumption is used to distinguish imports and domestically produced goods, implying imperfect substitutability and to differentiate exports from goods for domestic use. The production possibility frontier of the economy is defined by a constant elasticity of transformation (CET) function between domestic supply and export and this yields the export supply function. The model defines a composite commodity made up of domestic demand and final imports, which is consumed by the households, firms and government. We assume a constant elasticity of substitution (CES) function between domestic demand and final import demand and this provides the import demand function. Import and locally produced goods are subject to the same taxation structure. The parameters of CES and CET functions are exogenously determined.

Households receive income from primary factor payments that they hold directly, transfers from government and from the rest of the world. Households spend in the consumption of goods and services, pay taxation, and saved. Household demands are represented by a linear expenditure system (LES) derived from the maximization of a Stone-Geary utility function. The analysis of the distributional impact of tax reform is complemented by a microsimulation module in which demand functions are the same as those used in the CGE model.

Government revenues are generated of all levied taxes, i.e. direct taxes, production taxes, import duties, export duties, and indirect internal taxes (VAT and excise duties), as well as foreign aid and remittances from the rest of the world. As for their tax component, revenues depend on exogenous tax rates but also on the variation of their tax base. International transfers to the government are exogenous. The government produces public services, consumes various goods, and remunerates civil servants. Government savings are residual and correspond to government revenue from which are subtracted public expenditures and transfers from government to Nigerien households.

Prices are modelled in standard fashion. Nevertheless, this work highlights a few specificities of our modelling of prices such as the market price of goods and services integrating the cost of marketing margins and transport margins and their VAT rate. We will detail VAT treatment further on.

The equilibrium conditions and closure of the model are traditional. For each market, equilibrium is ensured by a price mechanism that endogenously adjusts the supply and demand. Investment is determined by the level of savings of agents. Household savings and public savings, for their part, are residual and endogenous. The nominal exchange rate is chosen as numeraire; transfers between the various agents and foreign saving are exogenous and fixed to their initial values. The balance of payment is balanced by the adjustment of the real exchange rate. Public expenditure is fixed, which is easily explained in a developing country such as Niger 12 and corresponds to our analytical frame distinguishing the effect of an increase in public revenues and the establishment of an effective policy for poverty reduction.

Specificities in the treatment of internal indirect taxation

The value added tax is an ad valorem tax applies theoretically to the price of the composite good (locally produced and imported goods) including trade and transport margins. In industry, VAT exemptions primarily concern the mining sector and firms that produce food, beverages, and tobacco. In services, it is foremost hotels and restaurants that are concerned. The proportion of VAT collected on intermediate consumption goods is fixed by sector, according to the available statistics. 14 It is 80% of total VAT revenues for food crop agriculture; 60% for export agriculture, and 30% for the sector of industry and services. The sectors that bear the greatest tax burden on intermediate consumption are in fact export agriculture and industry, and this is related to the level of input use in production. The burden of VAT collected on intermediate consumptions in the initial situation influences the results of our simulations when the pure VAT system is re-established and when the cascading effect is cancelled. With this distribution of collected VAT revenues, we obtain the VAT rates presented in Table 1. 12 Between 1999 and 2003, government spending remained stable between 17 and 18% of GDP. 13 Report: l'évaluation de l'application des directives de l'UEMOA, portant harmonisation des législations des Etats membres en matière de taxe sur la valeur ajoutée et de droits d'accises. 14 Information on the sum of exemptions is provided by the country's DGI for the year 2006. We have considered low substitution elasticities of foreign trade for all products. In other words, we assume in standard fashion that there is a significant differentiation between domestic, exported, and imported products (de Melo 1998). We have done the same for the substitution elasticity of production factors (Appendix 4).

Differential analysis of the social impacts of different VAT systems

To analyse the effect of the end of exemption, five simulations were run. The first two options (simulations 1a and 1b) take up one of the propositions for Nigerien economic policy, namely the application of a uniform rate and the broadening of the tax base to primary consumer goods. The theoretical objective of a single-rate VAT system is not a high rate but the largest possible tax base, thus distributing the tax burden over the economy while favouring optimal budget performance. This is made possible by exploiting tax loopholes, by a near-zero sum of exemptions and by the taxation of agriculture [START_REF] Bonjean | Le paradoxe de la fiscalité agricole en Afrique subsaharienne[END_REF]. These two first simulations capture the implementation of a pure VAT in the Nigerien economy in its theoretical dimension: no VAT on intermediate consumptions for any sectors expressing the total absence of VAT exemptions. In simulation 1a, we restore the official rate of 19% VAT imposed by the Finance act. In simulation 1b, we reduce the VAT rate at 10% as a result of the expansion of the tax base, which is consistent with the logic of the single rate. The model considers an endogenous public revenues and an exogenous VAT rate in order to assess the impact of a pure VAT on both public revenues and poverty.

The principle of "Equal Yield Tax Analysis" (Shoven and al., 1990) is made for the two following simulations. It aims to compare various taxation schemes while keeping constant the public expenditure and the State revenue. This analytical approach is adapted to capture the intrinsic incidence of alternative tax scenarios by isolating them from induced effects which may be related to the change in the State revenue, the State expenditures and savings. The public revenue in simulation 1b has been used as reference.

Simulation 2 takes into account the difficulty of taxing food crop agriculture products in developing economies. These goods are mostly exchanged in the informal sector or home consumed, which makes their consumption difficult to control and tax. The exemption of food crop agriculture implies maintenance of VAT cascading effect in the sector. This simulation should allow us to establish the net social impact of the end of exemptions in a sector where produced goods have been by far consumed but where the producer's vulnerability is also very important. Under the hypothesis of budget neutrality, VAT exemptions as well as the VAT rate in food crop agriculture are kept exogenous at their initial level but the VAT rate on final consumption of other sectors is endogenous and uniform.

In simulation 3, we decide to impose a 5% rate on the whole of food crop agriculture. Under the hypothesis of budget neutrality, the VAT rates in the other sectors are endogenous and unique. Government revenue is assumed to be the same as in simulation 1b. This simulation takes up the economic theory of multiple rates without taking into account realities tied to self-consumption and the informal sector, thus enabling a comparison with the theoretical system of the single rate. The VAT is fixed exogenously without budget neutrality in simulation 4. We have fixed a reduced VAT rate of 5% in the sector of food crop agriculture associated with a VAT rate of 15% in the other activity sectors.

The last two simulations (3 and 4) take into account the rigidity of a pure VAT system to test the hypothesis of the progressivity of a double-rate VAT system and thus analyse the social impacts of this taxation compared to the establishment of a single rate. Various authors have concluded that a single-rate consumption tax would be undesirable from an economic and social standpoint, as [START_REF] Gautier | Taxation optimale et réformes fiscales dans les PED: une revue de littérature tropicalisée[END_REF] shows in his literature review. The establishment of a reduced rate on a few sensitive products could provide a way to avoid the multiplication of ad hoc or permanent exemptions while avoiding penalizing the consumption of these products with a 19% or even 10% tax rate. It is important to note that this VAT system requires proper functioning of the tax authorities. The multiple rates can create VAT credits for firms when a part of the inputs is taxed at the full rate and the final product, at a reduced rate. To preserve neutrality of the VAT, all tax credits must be reimbursed by the tax authorities.

We will begin with a presentation of the sectoral and macroeconomic results before proceeding to a distributional analysis. The macroeconomic results of simulation 1b are detailed, as it corresponds to the reference simulation for the comparative analysis under the hypothesis of budget neutrality.

Macroeconomic results

Results of simulations 1a and 1b

In the two first simulations (1a and 1b), any possibility of imperfection related to the application of VAT have been eliminated (absence of informal sector, improved tax collection, absence of market power). VAT rate on final consumption increases considerably compared to the initial situation. The results of simulation 1b, will serve as the reference simulation for the subsequent comparative analysis. The results of simulation 1a are presented in Table 2 and these shed light on what would take place with the application of a pure VAT system associated with a high rate.

In this simulation (1b), two effects are combined. The first is related to the end of VAT When we integrate an official 10% VAT rate on the whole of goods and services, we see that the net effect is different than the one tied to the elimination of exemptions. On the local market, while the decrease in the domestic price resulted in a decrease in the price of the composite good permitting a decrease in the price of intermediate consumptions, the VAT rate strongly increases retail prices, which reduces domestic demand. Only the industrial good follows a different trend, as it benefits from a sufficient reduction in the price of the composite good for the VAT increase not to increase its retail price. Because industry is a driving force in the economy (production and investment), this amplifies the demand on this sector and its expansion. Given our modelling assumptions, the behaviour of the retail branch depends on the level of production in the rest of the economy. The decrease in the value added price leads to a decrease in the demand for labour, except in industry. It can be seen that the average wage does not change compared to the initial situation. The earned rate of return follows the same trend as the sectors' demand for labour. The variations of factor payments and factor allocation affects the level of disposable income of households and thus their welfare, the poverty level, and inequalities, which we will analyse in the distributional analysis.

In terms of public finance, the end of exemptions and the increase in the effective VAT rate yields a considerable increase in tax revenues. The government benefits from exemptions' suppression and from the re-establishment of a 10% effective VAT. VAT revenues now represent 11.70% of the GDP. Overall government revenue increases by 40.30%. The result in terms of public finance is nevertheless conditional upon the efficiency of the tax system.

The positive effect noted on public finance impacts the level of investment by an increase in value of 45.05% compared to the initial situation. This is primarily a result of government savings since the total savings of Nigerien households decrease.

The reduction in VAT exemptions allows the industrial sector to develop by increasing the domestic demand (intermediate consumption, investment) as well as its participation in foreign trade, but the size of the other sectors is reduced given the strong reduction in the local demand of households. It should be noted that with a single VAT rate of 10%, the macroeconomic variables only slightly vary compared to the initial situation (which is not the case with a 19% VAT rate) where public finance improves considerably.

Results of simulations 2 and 3

Under the hypothesis of budget neutrality, the major macroeconomic indicators are only slightly influenced by the different simulated VAT systems. The evolution of public finance, investment, average wage, and the price index remain similar in simulations 1b, 2, and 3.

Government revenues are fixed to the sum defined in simulation 1b. We should note that the variation in the utility of the representative agent remains similar.

In simulation 2, the VAT rate in sectors other than food crop agriculture is established at 12.86%. Preserving food crop agriculture as an informal sector thus requires a VAT increase on other goods. Maintaining food crop agriculture as a sector under taxation opens the way for supporting its development through a stabilization of domestic demand compared to the other strongly taxed goods. The value added price of food crop agriculture still decreases, as producers must now obtain intermediate consumptions at a 10% VAT rate without the possibility of having their VAT credits reimbursed. This produces stronger VAT cascading effect compared to the initial situation, in which the other sectors were also less strongly taxed. Nonetheless, the value added price of food crop agriculture decreases only slightly, as Nigerien farmers use very few inputs. Food crop agriculture can thus maintain its activity compared to the other sectors experiencing a slow-down in their development comparable to that of a pure VAT system (except for industry, cf. simulation 1b). The production factors move from export agriculture and services to industry, as in simulation 1b. Food crop agriculture maintains its production capacities. In this simulation, the additional cost produced by the VAT exemption is reduced by a decrease in the retail price on this good, which is not the case for the other sectors taxed at 10%. The price index increases slightly at 105.16 and total private consumption decreases in volume by 5.30% with a strong decrease for the consumption of services and a slight decrease for food crop agriculture.

When a dual VAT rate with a reduced rate is introduced in food crop agriculture at 5% and an endogenously fixed VAT rate on other goods at 11.37%, the macroeconomic situation comes closer to the results of simulation 1b but do not favour consumption of the other goods any more than in simulation 2.

Results of simulation 4

The introduction of a multiple-rate VAT structure with a 5% reduced rate in food crop agriculture and a 15% rate in the other sectors yields a 59.36% increase in public revenues compared to the initial situation, that is, 19 percentage points more than when we apply budget neutrality or with a single rate of 10%. Public savings, and hence investment, increase substantially. The VAT revenues now correspond to 15.1% of GDP, which is significantly higher than the WAEMU average and shows higher budget performance than with a 10% single rate. Household welfare nevertheless decreases by 10.51%, more than two points higher compared to the previous simulations. In terms of production, the situation is not very different from simulation 3.

On the whole, we observe that the macroeconomic effects of the different scenarios are very similar. However, it is important to analyse whether anticipations of the regressive nature of a pure VAT materialize through our distributional analysis. This analysis will also enable us to identify the best options in distributional terms.

Distributional analysis

The micro-simulation model associated with the CGE model enables us to study the effects of an economic policy for households observed in the survey. This technique takes into account the heterogeneity of households in terms of both their income structures and their expenditure structures observed in the household survey. This type of analysis is not possible with an approach based on representative agents. Moreover, according to Savard Regardless of the group, the average per capita expenditure of households in food crop agriculture remains the largest expenditure item, followed by the industrial good. Selfconsumption is included in food crop agriculture17 .The importance of expenditures in agricultural goods may justify the fact that government and donor agencies favour this expansion of the tax base. We observe that the difference in the consumption structure depends more on the area of residence than on the activity of the head of household. For urban households with or without agricultural activity, food crop agriculture exhibit lower share of expenditure and their consumption is more diversified. Industrial goods and services occupy a slightly less important place in their budget ( Table 3).

We observe a relation between obtaining revenue from agriculture and the level of poverty, irrespective of the area of residence. The group least affected by poverty is the urban nonfarmers (34%) while rural households headed by a farmer are most affected by poverty (70.8%). The Gini coefficient measuring the degree of inequalities has a value of 0.46 in 2005 in Niger (INS). Regardless of the group examined, the degree of inequalities is comparable despite a slightly higher rate among urban dwellers who do not obtain their revenue from agriculture (0.434). The expansion of the tax base to the whole of consumer goods and the re-establishment of a pure VAT system is costly in social terms (simulations 1a and 1b). If we consider simulation 1b, we can see that the most negatively affected group is the agricultural rural households (Ragr) with a 7.92% increase in poverty incidence. This group was initially the second poorest. The increase in the poverty depth and severity for this simulation is nevertheless greater for those in urban agricultural households (Uagr), that is, the poorest household group at the reference situation. It is also interesting that even if the effects on poverty are greater in terms of the impact on agricultural households residing in rural areas compared to those in urban areas, the variations in average revenue in these two groups are very similar (Appendix 5). This suggests that price effects play an important role for changes in poverty indices for rural agricultural households. We should note that this is true regardless of the 18 The results for each simulation and for each poverty index are available in Appendix 1. 19 This is consistent with negative variations in average revenues observed in Appendix 5. Considering a more realistic tax base by maintaining food crop agriculture under taxation is much less negative in terms of poverty variation (simulation 2). For a similar macroeconomic activity, the variation in the effect of poverty on rural agricultural households drops from 7.92% to 3.37%. The increase in the depth and severity indices is also much weaker. We should note that our results are significant only for the groups of rural farmers and nonagricultural urban groups.

Establishing a dual VAT rate by taxing food crop agriculture at a reduced rate of 5% (simulation 3) amplifies all poverty impact for all groups except for rural non-farmer, compared to the initial situation and compared to simulation 2. Indeed, poverty incidence increases by 5.83% for the group of rural agricultural households, i.e. 2.46 percentage points more than simulation 2. It should be noted that the negative social impact of this VAT system is nonetheless weaker with a single-rate VAT of 10%. When compared to uniform rate, allocating taxes on all goods increases poverty even when the full rate of this system (11.4%) is higher than the uniform rate (10%).

Finally, if we compare simulation 4 with simulation 1b without budget neutrality, we can see that putting into place a dual-rate VAT system will imply that the government will favour one group over another. The difference in terms of poverty increase primarily affects rural farmers. Maintaining the reduced rate in food crop agriculture, thus permitting support of agricultural activity and hence revenues, increases the poverty of this group in lesser proportions than the single rate of 10%. Urban non-farmers for their part are more affected by the dual rate, which follows the same logic.

The establishment of a reduced rate of 10% (simulation 1b) or multiple rates (simulation 3) reduce inequalities among rural households regardless of their activity, as well as among urban non-farm households. It is difficult to define what type of VAT system is preferable between simulation 1b and 3, despite the fact that the single rate allows a greater reduction in inequalities within the group of rural farmers. Simulation 2, maintaining VAT exemptions in food crop agriculture, does not provide significant information when it comes to the modification of inequalities, which prevents concluding on the benefits of this fiscal policy on this criteria ( Although restoring the VAT rate would be socially costly compared to the initial situation, we can see that the distributional impact of the VAT differs according to the system implemented in the country. As for the impact on poverty, we can conclude that the establishment of a single-rate VAT associated with maintaining exemptions in food crop agriculture is preferable for all social groups identified, since the negative effect proves to be the weakest. Moreover, the reduction observed in inequalities suggests that although the number under the poverty line would be higher, it may be possible to target a larger number of households during the establishment of compensatory measures favourable to the poor.

Conclusions

This study has sought to describe the distributional impacts of expanding the tax base to encompass agricultural goods by considering various potential taxation systems. To do so, we built the first CGEM micro-simulation for Niger and performed a rich distributional analysis of a VAT reform in a developing country using this tool. Three main lessons emerge from our application.

First, the end of VAT exemptions is necessary to develop the industrial sector, which has a strong ripple effect on the rest of the economy. Nonetheless, maintaining VAT exemptions in food crop agriculture sectors associated with a tax base expansion in the remaining sectors is preferable for the agriculture expansion. On the first hand, this fiscal 20 The complete results and significance for each simulation are available in table form in Appendix 6. choice avoids an increase in the tax burden on consumers which support demand. On the other hand, agriculture being a sector that uses little intermediate consumption, the tax burden tied to VAT exemptions is weak for the producer, which maintains the level of activity of this sector and hence the remuneration of farmers. The net distributional impact of VAT exemptions depends on the economic structure economic sectors and the initial tax burden on intermediate consumption. Furthermore, irrespective of the VAT system chosen, one must correctly apply the zero rate of VAT to avoid creating distortions between sales on the local market, where the VAT is neutral, and exportation.

Secondly, the model shows that applying a pure VAT structure conforming to the theory (simulation 1b) is preferable in terms of economic growth whereas applying a reduced-rate on food crop agriculture lightens the negative distributional impact of the elimination of exemptions compared to a single rate (simulation 4). The pure VAT structure induces an increase in poverty especially for rural farmers, the largest and second poorest group in Niger. Both simulations without imperfection in VAT application are border case.

Finally, when we take into account budget neutrality, we can see that maintaining VAT exemptions in food crop agriculture (simulation2) is the most judicious choice compared to a 5% VAT rate taxation of goods produced by this sector (simulation3). It is not the multiple rate VAT that is more progressive but the one with the tighter fiscal base It must be kept in mind that all of our results relative to the establishment of a single rate, multiple rates, or exemptions on certain goods depend on the ability of the Nigerien's tax authorities to control and reimburse VAT credits. To date, no study has evaluated the administrative cost of implementing the above-described VAT systems in developing or developed countries.

We intend to pursue this analysis by modifying certain hypotheses of the model and introducing different market structures in certain sectors of the economy. This will allow us to take into account the power of retailers and/or producers to determine market prices.

( )

Equilibrium equations:

∑ ∑ ∑ ∑ ∑ ∑

  Exemption is modelled as a supplementary tax paid by the producer during the purchase of intermediate consumptions. Because of the hypothesis of perfect competition, the producer cannot charge this tax burden on the sales price of the output. We have consequently modelled two types of VAT: one concerning the price of intermediate consumptions (TCI) when the final good is exempted and when VAT credit are not refund, and the other concerning the final retail price (TC). If the VAT rate affecting intermediate consumptions is positive, the price of intermediate consumptions increases for the producer and the value added price decreases. The VAT exemptions are thus taken into account and the VAT is not only a consumption tax VAT rate applied to final consumption and to intermediate consumptions is calculated at the base year according to the proportion of VAT revenues collected on intermediate consumption goods. The sum of VAT collected on intermediate goods in food crop agriculture can be explained by the large number of exemptions but also by the informality of the sector. In export agriculture, VAT collected on intermediate goods is due to the nonapplication of zero-rate. This repayment default concerning exporting firms exists in Niger. 13

  exemptions and the second, to the establishment of an effective VAT rate of 10%. The restoration of VAT neutrality in the production process leads to a decrease in the cost of intermediate consumptions, which increases the value added price, irrespective of sectors.While the value added increases in food crop agriculture and services-expanding sectorsit decreases in industry and export agriculture. The increase in value added is nonetheless not sufficient in these two sectors to compensate the increase in cost of the labour. There is a reallocation of the labour towards food crop agriculture and services. In terms of foreign trade, this simulation shows that from this point on, the same tax level applies for selling locally or exporting (the zero rate of VAT being correctly applied) and the decision is made solely according to relative prices. Contrary to the other sectors, domestic price in industry decreases, which favours exports and reduces the price for the producer (the largest share of industrial production is sold locally in the initial situation). This situation favours the purchase of locally produced goods. The domestic price for other goods including export agriculture increases. It penalizes local sales in favour of imported goods. The fix current account balance explains that increase in exports finances imports. The export agriculture and industry sectors experience an impact different from the other branches owing to the greater burden of VAT on their intermediate consumption modelled in the reference situation, but also given their positioning on the international market.

(

  2005), distributional analysis by representative agent can lead to qualitatively and quantitatively different conclusions. This type of model combining CGE and micro-simulation modules enables the distributional impact analysis of a tax policy on the real income of each household. Indeed, the model allows us to obtain new revenue and expenditure vectors after each simulation of economic policies implemented. New real income vector are computed after each simulation and thus it is possible to re-estimate poverty and inequality indices at the reference period and for each scenario.Our study draws on data from the 2005 Questionnaire Unifié sur les Indicateurs de Base du Bien-être (QUIBB) including 6 690 Nigerien households. The poverty analysis is performed using the usual Foster, Greer and Thorbecke (1984) poverty measures (FGT)15 . This covers poverty incidence (α=0), depth (α=1), and severity (α=2). The poverty line used is 105 827 CFAF per year per capita in rural areas and 144 750 CFAF in urban areas in 2005 (PNUD 2005). As for the inequality measure, we selected the Gini index.Four household groups were identified, first taking into account the place of residence: urban (households residing in Niamey and in other urban centres) and rural areas and second taking into account the main sector of activity of the head of household: agriculture or nonagriculture. The four groups analysed are thus Uagri, Ragri, Unagri, Ragri respectively representing 3.2%, 60%, 13.1%, and 23.7% of Nigerien households.16

Figure 1

 1 Figure118 presents the variations in the FGT indices for the four household groups and for the five simulations.

Figure 1 :

 1 Figure 1: Variation of household poverty indices (%)

  it is less marked in other cases. Households obtaining their income from a non-agricultural sector are less concerned. It can also be seen that reducing the VAT rate to 10% (simulation 1b) reduces the negative social repercussions of an expansion of the tax base compared to a 19% single rate (simulation 1a).

Figure

  Figure2).

Figure 2 :

 2 Figure 2: Variation in the Gini index (%) 20

Table 1 : Effective VAT rates in Niger according to sector and tax base

 1

	Sectors	TC rate	TCI rate
	Food crop
	agriculture	0.02	0.80
	Export agriculture	0.73	6.59
	Industry	7.50	7.72
	Service	0.04	0.05
	TC: VAT rate on final consumption, TCI: VAT rate on
	intermediate consumptions. Authors' calculations based
	on data from the SAM, 2004. The calculation method can
	be consulted in Appendix 2.

Table 2 : Macroeconomic results of the simulations (Δ %)

 2

	Selected variables	Sim1a	Sim1b	Sim2	Sim3	Sim4

Table 3 : Structure of average consumption among Nigerien households

 3

	Household groups	Uagr	Ragr	Unagr	Rnagr
	Food crop agriculture	43.66	56.53	38.26	52.40
	Export agriculture	16.48	14.33	16.94	15.37
	Industry	34.06	25.58	33.72	27.53
	Services	5.8	3.56	11.08	4.7
	Total	100	100	100	100
	Source: Authors' calculations based on data from the QUIBB_2005

Table 4 : Analysis of poverty and inequalities in Niger

 4

		Uagr	Ragr	Unagr	Rnagr
	Incidence (FGT0)	70.8	64.4	34.0	53.1
	Depth (FGT1)	30.3	25.8	10.8	19.4
	Severity (FGT2)	15.9	13.3	4.8	9.6
	Gini	0.427	0.429	0.434	0.428
	Authors' calculations based on data from the QUIBB_2005;

Directives No. 02/98CM/UEMOA on VAT and No. 03/98/CM/UEMOA on excise duties.

1999 survey on household consumption and data from the Ministère de l'économie et des finances, Institut National de la Statistique du Niger, (2008) Report « L'analyse de la politique fiscale au Niger et ses impacts sur la pauvreté ».

Exemption may be defines either in terms of particular commodities or in terms of particular trader.Only the first one is studied here.

In 2005, Niger experienced inflationary pressure notably owing to a strong reduction in agricultural production in 2004 due to climate problems.

According to[START_REF] Gautier | Les paradoxes de la TVA dans un pays en développement : l'illustration de l'industrie malgache[END_REF], "Net losses in government revenues are then equal to the sum of sales of exempted firms, from which are subtracted the sum of non-exempted inputs purchased by nontaxable branches, and exempted sales purchased by taxable firms."

[START_REF] Cockburn | Trade Liberalisation and Poverty in Nepal A Computable General Equilibrium Micro Simulation Analysis[END_REF] used micro-simulation to analyse the impact of a decrease in trade tax and en increase in consumption tax.

This sector includes the mining industry, food and beverage production, textile and clothing production, the chemical industry, metal production, other factories, electricity, gas, water, and finally construction.

The initial SAM accounted for the VAT collected in levy in import taxes without distinction, which strongly underestimated VAT revenues.

The distributional analysis was conducted using DASP (Distributive Analysis Stata Package) software.

It is important to point out that the decomposition of households is independent from the CGE model. It is carried out based on data from the QUIBB survey.

Self-consumption is more important in rural area. In the micro simulation model we have consider self-consumption as a capital income. Self-consumption and food crop agriculture can be combined due to their under taxed character.

Authors' calculations, results of the micro simulation on DASP. **significant for a threshold of 5%, *significant for a threshold of 10%.

Appendix

Appendix 1: Presentation of the sectors of the Nigerien economy Price block:


Tschauner-Hempel equations in Hill's frame

Keplerian motions -Small intersatellite distance (a, e, v ): semi-major axis, eccentricity and true anomaly of the target satellite

X (v ), Y (v ), Z (v ): relative coordinates of the chaser   x 1 (v ) x 2 (v ) x 3 (v )   = (1 + ecos(v ))   X (v ) Y (v ) Z (v )  
x 4 (v ), x 5 (v ), x 6 (v ): derivatives of x i (v ), (i = 1, ..., 3) w.r. The problem to solve

(P)                                    Find u = argmin u J(u) = -m(v f ) s.t. ẋ(v ) = A(v )x(v ) + B(v ) u(v ) m(v ) ṁ(v ) = -c(v ) u(v ) u(v ) ≤ 1 v ∈ [v 0 , v f ] g (v , x(v )) ≤ 0 v ∈ [v 0 , v f ] x(v 0 ) = x 0 h (x(v f )) = 0 m(v 0 ) = m 0 Richard Epenoy Fuel-Optimal
g (v , x(v )) = 1 - x 1 (v ) 2 + x 2 (v ) 2 + x 3 (v ) 2 d min (1 + ecos(v )) ≤ 0 v ∈ [v 0 , v f ]

Problem regularization (1/2)

A logarithmic barrier approach The smoothed exact penalty method

(P) δ                                  Find u δ = argmin u J δ (u) = J(u) -δ v f v 0 F (v , u(v )) dv s.t. ẋ(v ) = A(v )x(v ) + B(v ) u(v ) m(v ) ṁ(v ) = -c(v ) u(v ) u(v ) ≤ 1 v ∈ [v 0 , v f ] g (v , x(v )) ≤ 0 v ∈ [v 0 , v f ] x(v 0 ) = x 0 h (x(v f )) = 0 m(v 0 ) = m 0 F (v , u(v )) = c(v ) (log ( u(v ) ) + log (1 -u(v ) ))
J δ (u δ ) → J(u) as δ → 0 u δ → u for the weak-* topology on L ∞ [v 0 , v f ] , R 3 as δ → 0 Richard Epenoy Fuel-Optimal
The penalized problem (P) The smoothed exact penalty method

2 σ,α,                              Find u 2 σ,α, = argmin u J 2 σ,α, (u) = J δ (u) + v f v 0 ψ σ,α, (g (v , x(v ))) dv s.t. ẋ(v ) = A(v )x(v ) + B(v ) u(v ) m(v ) ṁ(v ) = -c(v ) u(v ) u(v ) ≤ 1 v ∈ [v 0 , v f ] x(v 0 ) = x 0 h (x(v f )) = 0 m(v 0 ) = m 0 ψσ,α, (z) = σlog 1 + exp z σ 1 + α -z -→ σ→0 Max 0, z 1 + α -z
The penalized problem (P) 2 σ,α,

                             Find u 2 σ,α, = argmin u J 2 σ,α, (u) = J δ (u) + v f v 0 ψ σ,α, (g (v , x(v ))) dv s.t. ẋ(v ) = A(v )x(v ) + B(v ) u(v ) m(v ) ṁ(v ) = -c(v ) u(v ) u(v ) ≤ 1 v ∈ [v 0 , v f ] x(v 0 ) = x 0 h (x(v f )) = 0 m(v 0 ) = m 0 G. Liuzzi
Algorithm Let 0 < q 1 < q 2 < 1, 0 < α lim < α 0 , 0 > 0, σ 0 > α q 1 lim , 0 < θ < 1, 0 < τ < 1 Let k = 0, end = false WHILE (end = false) Solve problem (P) 2 σ k ,α k , k → x 2 σ k ,α k , k , m 2 σ k ,α k , k , u 2 σ k ,α k , k IF (α k ≤ α lim ) THEN end = true (x δ , m δ , u δ ) ← x 2 σ k ,α k , k , m 2 σ k ,α k , k , u 2 σ k ,α k , k ELSE IF Min k , v f v 0 Max 0, g (v , x 2 σ k ,α k , k (v )) dv > α q 2 k σ k THEN k+1 = τ α q 2 k σ k ELSE k+1 = k ENDIF α k+1 = θα k σ k+1 = Min σ k , α q 1 k+1 k = k + 1 ENDIF END WHILE

Convergence results

Lemma Let α > 0, z < α and > 0 be given. Then, (σ → ψ σ,α, (z)) is strictly increasing on ]0, +∞[ Let z < 0, > 0 and σ > 0 be given. Then, (α → ψ σ,α, (z)) is strictly increasing on ]0, +∞[ Convergence theorem -See Epenoy (2011) in JGCD

J 2 σ k ,α k , k u 2 σ k ,α k , k → J δ (u δ ) as k → ∞ u 2 σ k ,α k , k → u δ according to the weak-* topology on L ∞ [v 0 , v f ] , R 3 as k → ∞ x 2 σ k ,α k , k → x δ uniformly on [v 0 , v f ] as k → ∞ m 2 σ k ,α k , k → m δ uniformly on [v 0 , v f ] as k → ∞
= 0.0 s v f = 3.349161118514 rad t f = 8000.0 s   X (t 0 ) Y (t 0 ) Z (t 0 )   =   -100 m -100 m -100 m          dX dt (t 0 ) dY dt (t 0 ) dZ dt (t 0 )        =   0.0 m/s 0.0 m/s 0.0 m/s     X (t f ) Y (t f ) Z (t f )   =   500 m 500 m 500 m          dX dt (t f ) dY dt (t f ) dZ dt (t f )        =   0.0 m/

  . Epenoy: Fuel Optimization for Continuous-Thrust Orbital Rendezvous with Collision Avoidance Constraint, Journal of Guidance, Control and Dynamics, Vol. 34, No. 2, March-April 2011. Epenoy Fuel-Optimal Trajectories for Path-Constrained Rendezvous Problem statement Solving the path-constrained rendezvous problem Numerical results -A rendezvous in Highly Elliptical Orbit Conclusion and future prospects Future prospects Minimum-fuel rendezvous in perturbed environment Necessity to modify the dynamical equations Low Earth Orbit applications =⇒ J 2 term of the Earth's potential Application to other state-constrained problems Reentry trajectories under thermal flux constraint Interplanetary trajectories with minimum flyby altitude constraint Toward closing the loop Model Predictive Control methodology Neighboring extremal paths in the presence of state constraints Hamilton-Jacobi-Bellman equation for state-constrained problems Richard Epenoy Fuel-Optimal Trajectories for Path-Constrained Rendezvous Problem statement Solving the path-constrained rendezvous problem Numerical results -A rendezvous in Highly Elliptical Orbit Conclusion and future prospects Thank you for your attention

	Problem statement Solving the path-constrained rendezvous problem Numerical results -A rendezvous in Highly Elliptical Orbit Conclusion and future prospects Problem statement Solving the path-constrained rendezvous problem Solving the path-constrained rendezvous problem Numerical results -A rendezvous in Highly Elliptical Orbit Numerical results -A rendezvous in Highly Elliptical Orbit Conclusion and future prospects Conclusion and future prospects	Statement of the test case Unconstrained rendezvous Rendezvous under collision avoidance constraint
	Numerical results -A rendezvous in Highly Elliptical Orbit Conclusion and future prospects Conclusion
	Unconstrained rendezvous
	1 Problem statement Problem statement The smoothed exact penalty approach
	Dynamical equations Dynamical equations Applied for the first time to an optimal control problem
	Optimal control formulation Optimal control formulation Efficient to deal with the collision avoidance constraint
	Solving the path-constrained rendezvous problem Implies solving a sequence of state constraint-free problems
	Smoothing the bang-off-bang control More theoretically based than other penalization techniques A new approach to deal with the state constraint Can be used to solve a large class of problems
	Numerical results -A rendezvous in Highly Elliptical Orbit
	Statement of the test case A just published paper
	Unconstrained rendezvous		s	
	Rendezvous under collision avoidance constraint	0.0 m/s	
	Conclusion and future prospects		0.0 m/s
	Richard Epenoy Richard Epenoy Richard	Fuel-Optimal Trajectories for Path-Constrained Rendezvous Fuel-Optimal Trajectories for Path-Constrained Rendezvous

R

Richard Epenoy

Fuel-Optimal Trajectories for Path-Constrained Rendezvous

Solving the path-constrained rendezvous problem Smoothing the bang-off-bang control

Parameters of the algorithm

Smoothing parameter: The same as for the unconstrained problem

Case 1: d min = 50.0 m q 1 = 0.3 q 2 = 0.6 α lim = 4.73x10 -4 α 0 = 0.98 0 = 0.0999 σ 0 = 1.0 θ = 0.99 τ = 0.9

Case 2: d min = 140.0 m q 1 = 0.3 q 2 = 0.6 α lim = 1.37x10 -4 α 0 = 0.98 0 = 0.0157 σ 0 = 1.0 θ = 0.99 τ = 0.9


Thang Dinh

Alexandre Vu

Rémy Renaux

Sylvie Boyer

Marcos

WEISS-WEINSTEIN BOUND AND SNR THRESHOLD ANALYSIS FOR DOA ESTIMATION WITH A COLD ARRAY

Keywords: DOA and polarization estimation, Weiss-Weinstein bound, COLD array

In the context of polarized sources localization using a cocentered orthogonal loop and dipole array, direction-of-arrival estimation performance in terms of mean square error are investigated. In order to evaluate these performance for both asymptotic and non-asymptotic scenarios (low number of snapshot and/or low signal to noise ratio) we derive closed-form expressions of the Weiss-Weinstein bound. The analysis is performed under both conditional and unconditional source signal models. We show the good ability of the proposed bound to predict the well known threshold effect. We also show the influence of the polarization parameters.

INTRODUCTION

In array processing, the use of an antenna able to handle the source polarization is of importance for a large class of application (see [START_REF] Li | Efficient direction and polarization estimation with a COLD array[END_REF] and references therein). One of such antenna is the so-called COLD (cocentered orthogonal loop and dipole) array. Several papers related to the COLD array can be cited: in [START_REF] Li | Efficient direction and polarization estimation with a COLD array[END_REF], the authors have introduced an efficient Direction-Of-Arrival (DOA) and polarization estimation technique and the COLD array was shown to improve both the DOA and the polarization estimation performance compared to the cocentered crossed-dipole array. Another example concerning asymptotic performance is given in [START_REF] El Korso | Statisti-cal resolution limit of the uniform linear cocentered orthogonal loop and dipole array[END_REF] where the statistical resolution limit, under the conditional observation model (i.e. when the source signals are assumed to be deterministic) was investigated. The authors have introduced an approach based on the Cramér-Rao bound (CRB), and again, under some conditions, the COLD array was shown to provide better performance than a conventional array.

In this paper, we are interested by the ultimate performance of such an array in non-asymptotic scenarios, i.e., when the Signal to Noise Ratio (SNR) and/or the number of snapshots are low. Indeed, it is well known that such an array processing problem is a nonlinear estimation problem for which a threshold phenomenon (i.e., when a drastic increase of the DOA estimator mean square error appears) [START_REF] Vantrees | Bayesian bounds for parameter estimation and nonlinear filtering/tracking[END_REF]. Unfortunately, the CRB can not capture this threshold effect [START_REF] Boyer | Analysis of the cold uniform linear array[END_REF]. Therefore, we are herein interested to investigate a more relevant bound for the COLD array observation model: the so-called Weiss-Weinstein bound (WWB) [START_REF] Weinstein | A general class of lower bounds in parameter estimation[END_REF]. Moreover, since the WWB is a Bayesian bound, it takes into account the support of the parameter via the prior distribution of the parameters. Consequently, the WWB provides a powerful tool to predict the global MSE behavior in both asymptotic and non-asymptotic regions. This project is funded by both the Digiteo Research Park and the Region Ile-de-France.

Due to its complexity (in comparison with the CRB), there are few publications related to the WWB in the literature. Some of previous works, such as [START_REF] Nohara | Application of the Weiss-Weinstein bound to a two dimensional antenna array[END_REF], have evaluated the WWB only by way of simulations (i.e., without closed-form expressions). The bound has been compared to the MSE of the MUSIC algorithm and classical Beamforming using an 8 × 8 element array antenna. In [START_REF] Nguyen | Comparison of performance bounds for DOA estimation[END_REF], the authors have introduced a numerical comparison between the Bayesian CRB, the Ziv-Zakai bound and the WWB for DOA estimation. In [START_REF] Athley | Optimization of element positions for direction finding with sparse arrays[END_REF], numerical simulations of the WWB to optimize sensor positions for non-uniform linear arrays have been presented. In [START_REF] Xu | Bayesian bounds for matched-field parameter estimation[END_REF], by considering the matched-field estimation problem, the authors have derived a semi closed-form expression of the WWB in the socalled unconditional observation model (i.e. when the source signals are assumed to be Gaussian). Concerning the aforementioned context of the conditional observation model a closed-form expression of the WWB is given in the simple case of spectral analysis in [START_REF] Renaux | Weiss-Weinstein bound for data aided carrier estimation[END_REF]. Recently, the closed-form expressions of the WWB for DOA estimation with a classical planar array were derived in [START_REF] Vu | Closed-form expression of the Weiss-Weinstein bound for 3D source localization: the conditional case[END_REF], and were applied for the array geometry design. Note that all these works have been performed in the context where arrays are not able to handle the polarization of the source signals.

In this paper, we consider the WWB performance bound on the DOA and the SNR threshold analysis of a polarized source with a COLD array. The closed-form expressions of the WWB are given under both conditional and unconditional observation models for which we present simulation results.

MODEL SETUP

We consider the context of DOA estimation of a single polarized source using a linear (possibly non-uniform) array of N COLD sensors. The source signal is assumed to be narrow-band and to be located in the far-field area. The sensor positions with respect to a reference axis are characterized by the vector d = [d1 . . . dN ]. As in [START_REF] Li | Efficient direction and polarization estimation with a COLD array[END_REF][START_REF] Li | Direction and polarization estimation using arrays with small loops and short depoles[END_REF] we assume that the source is coplanar with the array. Therefore, the DOA of the source signal depends only on the azimuth angle, denoted φ. We assume that the source polarization is known or previously estimated. Let us set u = j2πA sl λ cos ρ -L sd sin ρe (jψ) T , the polarization vector, with polarization angles ρ ∈ [0, π/2] and ψ ∈ [-π, π]. λ denotes the wavelength, and A sl , L sd denote the dipole lengths and the loop perimeters, respectively, with the assumption that A sl < 3λ/10, and L sd < λ/10. The output at the i th sensor and for the t th snapshot is a two components vector given by [START_REF] Boyer | Analysis of the cold uniform linear array[END_REF]:

ŷi(t) yi(t) = [a(φ)]is(t)u + ni(t), t = 1, . . . , T, (1)
where s(t) denotes the source signal, where [a(φ)]i = e (j 2π

λ d i sin φ)
denotes the i th element of the steering vector a(φ) and where ni(t) is an additive noise. This noise is assumed to be complex, circular, uncorrelated (spatially, temporally and between the loop and the dipole of each sensors) Gaussian with zero mean and the variance σ 2 . We will use the two classical alternative hypothesis about the source signal

• H1: the source signal is assumed to be deterministic and known. This is the so-called conditional signal model [START_REF] Ottersten | Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing[END_REF]. • H2: the source signal is assumed to be a complex circular random Gaussian vector, independent of the noise, with zero mean and known covariance matrix σ 2 s I. This is the so-called unconditional signal model [START_REF] Ottersten | Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing[END_REF].

For mathematical convenience, we consider the estimation of ω = sin φ, and we assume that ω follows a uniform distribution

ω ∼ U[-1, 1], i.e. p(ω) = 1 2 if -1 ≤ ω ≤ 1, 0 otherwise. ( 2
)
The output signal for the t th snapshot of the whole array can be then expressed as:

y(t) = ŷ(t) y(t) = b(ω)s(t) + n(t), (3)
where b(ω) = u ⊗ a(ω), with ⊗ denotes the Kronecker product, where ŷ(t) = [ŷ1(t) . . . ŷN (t)] T , and where y(t) = [y1(t) . . . yN (t)] T . With the aforementioned assumptions, under H1, the full observation vector (∀t) follows a Gaussian distribution with parameterized mean, i.e., y|ω ∼ CN (IT ⊗ b(ω)s, σ 2 I2NT ), with s = [s(1) . . . s(T )] T . And, under H2, y(t) follows a Gaussian distribution with parameterized covariance matrix, i.e., y|ω ∼ CN (0, IT ⊗ R(ω)), where R(ω) = σ 2 s b(ω)b(ω) H + σ 2 I2N Consequently, the likelihood function under H1 can be expressed as:

p(y|ω) = 1 (πσ 2 ) 2NT exp - 1 σ 2 T t=1 y(t) -b(ω)s(t) 2 , (4)
and the likelihood function under H2 is given by

p(y|ω) = 1 π 2NT |R(ω)| T exp - T t=1 y(t) H R(ω) -1 y(t) , (5)
where |.| denotes the matrix determinant and . 2 denotes the norm operator. The Weiss-Weinstein bound will be derived under both models H1 and H2.

WEISS-WEINSTEIN BOUNDS FOR THE COLD ARRAY

The Weiss-Weinstein bound is a lower bound on the mean square error well known to accurately predict the SNR threshold effect which appears in non-linear estimation problems [START_REF] Weinstein | A general class of lower bounds in parameter estimation[END_REF]. The WWB is generally obtained by taking the supremum of a function over a set of test point and over a set of parameter s ∈ [0, 1]. Concerning the parameter s, one often set s = 1/2 , see [START_REF] Xu | Bayesian bounds for matched-field parameter estimation[END_REF][START_REF] Renaux | Weiss-Weinstein bound for data aided carrier estimation[END_REF][START_REF] Vu | Closed-form expression of the Weiss-Weinstein bound for 3D source localization: the conditional case[END_REF], although there are no proof that s = 1/2 could leads to the tightest WWB. The WWB for s = 1/2 is given by:

W W B = sup h h 2 η(h, 0)η(0, h) 2(η(h, h) -η(h, -h)) ( 6
)
where h is the difference between the parameter of interest and a test-point and where the function η is defined as

η(α, β) = Θ Ω p(y, ω + α)p(y, ω + β)dydω = Θ p(ω + α)p(ω + β)ζ(α, β)dω, (7)
by denoting ζ(α, β) = Ω p( y| ω + α)p( y| ω + β)dy. Ω and Θ are the observation space and the parameter space, respectively. p(y, .) and p(.) denote the joint distribution of the full observation vector and the parameter (possibly a test point) and the a priori distribution of the parameter (possibly a test point). Note that one has to respect ω + h ∈ Θ.

Conditional observation model

From ( 4), the expression of ζ(α, β) is given by:

ζ(α, β) = Ω 1 (πσ 2 ) 2NT ×e -1 2σ 2 T t=1 ( y(t)-b(ω+α)s(t) 2 + y(t)-b(ω+β)s(t) 2 ) dy. ( 8
)
By substituting

x(t) = y(t)-1 2 (b(ω + α)s(t) + b(ω + β)s(t)), it easily leads to -1 2σ 2 T t=1 y(t) -b(ω + α)s(t) 2 + y(t) -b(ω + β)s(t) 2 = -1 σ 2 T t=1 x 2 + 1 4 b(ω + α) -b(ω + β) 2 (9) Since Ω 1 (πσ 2 ) 2NT exp - 1 σ 2 x 2 dx = 1, (10)
one obtains

ζ(α, β) = exp - s 2 4σ 2 b(ω + α) -b(ω + β) 2 . ( 11
)
Since b(ω) = u ⊗ a(ω), and since u H u =

4π 2 A 2 sl λ 2 cos 2 ρ + L 2 sd sin 2 ρ, the closed-form expression of b(ω + α) -b(ω + β) 2 is obtained by noting that b(ω + α) 2 = b(ω + β) 2 = N 4π 2 A 2 sl λ 2 cos 2 ρ + L 2 sd sin 2 ρ , (12) b
(ω + α) H b(ω + β) = 4π 2 A 2 sl λ 2 cos 2 ρ + L 2 sd sin 2 ρ × N i=1 e (j 2π λ d k (β-α)) , (13)
and that

b(ω + β) H b(ω + α) = 4π 2 A 2 sl λ 2 cos 2 ρ + L 2 sd sin 2 ρ × N i=1 e (j 2π λ d k (α-β)) . ( 14
)
One observes that the functions ζ(α, β) are no longer depending on the parameter ω, consequently, η(α, β) can be rewritten as:

η(α, β) = ζ(α, β) Θ p(ω + α)p(ω + β)dω. ( 15
)
Under the uniform distribution prior assumption, one obtains:

Θ p(ω + α)p(ω + β) = 1 - |α| + |β| 2 . ( 16
)
From ( 6), ( 12), ( 13), ( 14), ( 15) and ( 16), one obtains the closedexpression of WWB as (17) (shown on the top of next page). Note that the bound is independent of the parameter ψ.

Unconditional observation model

From ( 5), the expression of ζ(α, β) can be expressed as:

ζ(α, β) = Ω 1 π 2NT |R(ω + α)| T /2 |R(ω + β)| T /2 ×e - T t=1 y(t) H R(ω+α) -1 +R(ω+β) -1 2 y(t) dy (18)
By setting

Γ -1 = R(ω+α) -1 +R(ω+β) -1 2 , one obtains, |Γ| = 2 2N |R(ω+α) -1 +R(ω+β) -1 | , which leads to ζ(α, β) = |Γ| T |R(ω+α)| T /2 |R(ω+β)| T /2 × Ω 1 π 2N T |Γ| T exp - T t=1 y(t) H Γ -1 y(t) dy. (19)
Since

Ω 1 π 2N T |Γ| T exp - T t=1 y(t) H Γ -1 y(t) dy = 1, one ob- tains ζ(α, β) = |Γ| T |R(ω + α)| T /2 |R(ω + β)| T /2 . ( 20
)
Due to the special structure of R(ω +δ) = σ 2 s b(ω +δ)b(ω +δ) H + σ 2 I2N , one easily gets

|R(ω + δ)| = σ 4N 1 + σ 2 s σ 2 b(ω + δ) 2 . (21)
Furthermore, thanks to Woodbury identity, one gets

R(ω + δ) -1 = 1 σ 2 I2N - σ 2 s b(ω + δ)b(ω + δ) H σ 2 s b(ω + δ) 2 + σ 2 , (22)
thus

R(ω + α) -1 + R(ω + β) -1 = 1 σ 2 2I2N - σ 2 s b(ω+α)b(ω+α) H σ 2 s b(ω+α) 2 +σ 2 - σ 2 s b(ω+β)b(ω+β) H σ 2 s b(ω+β) 2 +σ 2 . ( 23
)
The matrix determinant of R(ω + α) -1 + R(ω + β) -1 equals to the product of their eigenvalues. Particularly, there are 2N -2 eigenvalues equal to 2/σ 2 , and the eigenvectors corresponding to the two remaining eigenvalues is a linear combination of the form b(ω + α) + qb(ω + β). Furthermore, the two above eigenvalues ν are the solution of the following equation

R(ω + α) -1 + R(ω + β) -1 (b(ω + α) + qb(ω + β)) = ν (b(ω + α) + qb(ω + β)) , (24)
(ω + α) 1 σ 2 2 -A b(ω + α) 2 -qAC -ν +b(ω + β) 1 σ 2 2q -Bq b(ω + β) 2 -BC H -qν = 0, ( 25
) where A = σ 2 s σ 2 s b(ω+α) 2 +σ 2 , B = σ 2 s σ 2 s b(ω+β) 2 +σ 2 and C = b(ω + α) H b(ω + β)
. Since b(ω + α) and b(ω + β) are nonlinear, thus, their coefficients are equal to 0. Solving the first coefficient for q and then substituting q into the second coefficient one obtains the equation

ν 2 σ 4 + νσ 2 2 -A b(ω + α) 2 -2 + B b(ω + β) 2 -4 + 2A b(ω + α) 2 + 2B b(ω + β) 2 -AB b(ω + α) 2 b(ω + β) 2 + ABCC H = 0. (26)
Solving (26) for ν, and since b(ω

+ α) 2 = b(ω + β) 2 = b(ω) 2 , one obtains R(ω + α) -1 + R(ω + β) -1 = 2N i=1 νi = 2 2N σ 4N σ 2 b(ω) 2 σ 2 s +σ 2 + 1 4 σ 4 s ( b(ω) 4 -C 2 ) ( b(ω) 2 σ 2 s +σ 2 ) 2 . (27)
Finally, substituting (21), ( 27) into (20), we have

ζ(α, β) = 1 + σ 2 s ( b(ω) 4 -b(ω + α) H b(ω + β) 2 ) 4σ 2 ( b(ω) 2 σ 2 s + σ 2 ) -T
.

(28) From ( 12), [START_REF] Ottersten | Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing[END_REF], and (14), one observes that, as for the conditional case, ζ(α, β) does not depends on the parameter ω. Consequently, the closed-form expression of the WWB is given by (29) (shown on the top of next page). Note that the bound is independent of the parameter ψ.

SIMULATION RESULT

In a first simulation, we compare the MSE behavior of the maximum a posteriori estimator (MAP) versus the WWB. We use an uniform linear COLD array consisting of N = 10 sensors with the intersensors spacing equal to λ/2 and T = 20 snapshots. The polarization parameter are set to ρ = π/4, ψ = π/3, A sl = 2λ/10, and L sd = λ/10. The empirical MSE of the MAP is obtained over 1000 On the other hand, the impact of the polarization parameter, ρ, is investigated. The scenario is the same as the previous simulation. Fig. 2 shows the WWB versus the SNR, according to different value of ρ under both assumptions. One observes that in each cases, both the SNR threshold and asymptotic MSE are affected by ρ.

W W B = sup h h 2 1 -|h| 2 2 exp -s 2 σ 2 4π 2 A 2 sl λ 2 cos 2 ρ + L 2 sd sin 2 ρ N - N k=1 cos 2π λ d k h 2 1 -|h| 2 -2 (1 -|h|) exp -s 2 2σ 2 4π 2 A 2 sl λ 2 cos 2 ρ + L 2 sd sin 2 ρ N - N k=1 cos 4π λ d k h . ( 17

Fig. 1 .

 1 Fig. 1. MAP versus WWB.

Fig. 2 .

 2 Fig. 2. WWB w.r.t the parameter ρ

CONCLUSION

In this paper, we have derived closed-expressions of the WWB in the context of source localization with a COLD array under both conditional and unconditional observation models. The WWB is shown to be a useful tool to capture the threshold effect. Furthermore, we showed that the polarization parameter ρ has a strongly impact on the MSE behavior.


Thang Dinh

Mohammed Vu

E L Nabil

Rémy Korso

Alexandre Boyer

Sylvie Renaux

Marcos

ANGULAR RESOLUTION LIMIT FOR VECTOR-SENSOR ARRAYS: DETECTION AND INFORMATION THEORY APPROACHES

Keywords: Angular resolution limit, polarized source localization, distance measure, information theory, detection theory

The Angular Resolution Limit (ARL) on resolving two closely spaced polarized sources using vector-sensor arrays is considered in this paper. The proposed method is based on the information theory. In particular, the Stein's lemma provides, asymptotically, a link between the probability of false alarm and the relative entropy between two hypothesis of a given statistical binary test. We show that the relative entropy can be approximated by a quadratic function in the ARL. This property allows us to derive and analyze a closed-form expression of the ARL. To illustrate the interest of our approach the ARL, in the sense of the detection theory, is also derived. Finally, we show that the ARL is only sensitive to the norm of the polarization state vector and not to the particular values of the polarization parameters.

INTRODUCTION

The Direction-Of-Arrivals (DOA) estimation for polarized sources based on the vector-sensor arrays have been largely investigated in the last decade. In [START_REF] Rahamim | Source localization using vector sensor array in a multipath environment[END_REF], the authors considered the problem of source localization in a multipath environment by using a vector-sensor array consisting of three electric and three magnetic dipoles. In [START_REF] Miron | Vector-sensor MUSIC for polarized seismic sources localization[END_REF], a tensorial version of the MUSIC algorithm for vector-sensor arrays was presented. On the other hand, the polarized seismic wave estimation was considered in [START_REF] Anderson | Analysis of a polarized seismic wave model[END_REF]. Whereas, in [START_REF] Nehorai | Vector-sensor array processing for electromagnetic source localization[END_REF], closed form expressions of Cramér-Rao Bound (CRB) have been derived.

In array processing, the Angular Resolution Limit (ARL) characterizes the minimum angular separation to resolve two closely spaced sources. In the literature, there are three approaches to obtain the ARL. [START_REF] Rahamim | Source localization using vector sensor array in a multipath environment[END_REF] The first approach based on the estimation accuracy. In this way, Smith [START_REF] Smith | Statistical resolution limits and the complexified Cramér-Rao bound[END_REF] proposed the following criterion based on the CRB: two signals are resolvable if the separation between the two DOAs θ1 and θ2 is less than the standard deviation of the separation estimation. Consequently, the ARL in the Smith sense is defined as the angular separation between the parameters of interest that is equal to the standard deviation of the angular separation. Furthermore, in [START_REF] El Korso | Statistical resolution limits for multiple parameters of interest and for multiple signals[END_REF], the extension of the ARL in the case of multiple parameters per signal based on Smith criteria was presented. [START_REF] Miron | Vector-sensor MUSIC for polarized seismic sources localization[END_REF] The second approach based on the concept of the mean null spectrum [START_REF] Sharman | Resolving power of signal subspace methods for fnite data lengths[END_REF]. This approach is quite intuitive but is only relevant to a specific high-resolution algorithm. (3) The third approach based on This project is funded by both the Digiteo Research Park and the Region Ile-de-France. detection theory. In [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF], the ARL based on the Minimum Probability of Error (MPE) for the deterministic signals is investigated. The authors used the first order Taylor expansion of the MPE to derive the ARL. Whereas, in [START_REF] Liu | Statistical angular resolution limit for point sources[END_REF] Liu and Nehorai have defined the statistical angular resolution limit using the asymptotic equivalence of the Generalized Likelihood Ratio Test (GLRT). On the other hand, Sharman and Milanfard [START_REF] Shahram | Imaging below the diffraction limit: A statistical analysis[END_REF] derived the frequency resolution limit in the spectral analysis using the GLRT.

In this paper, we propose a new approach of the ARL based on the information theory, and more specifically on the Stein's lemma [START_REF] Cover | Elements of information theory[END_REF]. The Stein's lemma links the false alarm probability (P f a ) resulted from the Neyman-Pearson decision criterion to the relative entropy (also called Kullback-Leiber pseudo-distance). As the relative entropy can be approximated by a quadratic function in the ARL, it is possible to determine the ARL by this way. To illustrate our approach, the ARL is derived in the Neyman-Pearson sense in the context of the Bayesian detection theory. Finally, we also compare our approach to the one presented by Amar and Weiss in reference [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF].

MODEL SETUP

Polarized signal model

We consider the context of DOA estimation of two narrow-band polarized source signals using a linear two elements vector-sensor array. Without loss of generality, we assume that the array lays on the Ox axis of the Cartesian coordinate. The array consists of N vector-sensors and the known positions of these vector-sensors in the array are given by the vector d = [d1 . . . dN ] T . The two sources are assumed to be located in the far-field, deterministic, and coplanar with the array, i.e., the source elevations equal θ k = π/2, k = {1, 2} 2 . For mathematical convenience, we consider the estimation of u k = 2π λ sin φ k , where λ denotes the wavelength. We assume that the source polarization is constant in time and along the array. The polarization of the sources is characterized by the vector [START_REF] Miron | Vector-sensor MUSIC for polarized seismic sources localization[END_REF]

p(ρ, ϕ) = [p(ρ, ϕ)] 1 [p(ρ, ϕ)] 2 = 1 ρ 2 + 1 1 ρe jϕ , (1)
where ρ and ϕ denote the amplitude ratio and the phase shift between the second component of the sensor and the first [START_REF] Miron | Vector-sensor MUSIC for polarized seismic sources localization[END_REF]. At the t th snapshot, the output signal in the time domain at the i th vectorsensor consists of two components (the model in frequency domain is available in [START_REF] Miron | Vector-sensor MUSIC for polarized seismic sources localization[END_REF])

ẑi(t) = 2 k=1 [p(ρ, ϕ)] 1 [a(u k )]is k (t) + ni(t) (2) zi(t) = 2 k=1 [p(ρ, ϕ)] 2 [a(u k )]is k (t) + ni(t) (3)
where a(u k ) = e -jd1 u k . . . e -jd N u k T , and where n(t), n(t) denote the additive noises at the t th snapshot, and s k (t) denotes the source signal of the k th source at the t th observation. We assume that s1(t) = s2(t), ∀t. The noises are assumed to be complex, circular, white Gaussian with zero mean and covariance matrix σ 2 I, i.e., n ∼ CN (0, σ 2 I). Furthermore, we assumed the signal sequences of the two sources are deterministic with known sequences at the observer. Consequently, at the t th snapshot, the response vector z(t) = [ẑ1(t) . . . ẑN (t) z1(t) . . . zN (t)] T of the array is given by

z(t) = 2 k=1 b(u k )s k (t) + n(t) (4)
where n(t) denotes the noise vector and b(u k ) = p(ρ, ϕ) ⊗ a(u k ) in which ⊗ stands for the Kronecker product.

Bayesian MPE based binary hypothesis test

We can now adopt the two detection hypotheses (see [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF]): under hypothesis H0, the observer detects only a single signal, which is a combination of the two sources, and under H1, the observer detects two signals:

   H0 : z(t) = b(û(t))ŝ(t) + H1 : z(t) = 2 k=1 b(u k )s k (t) + n(t) (5)
where û(t) and ŝ(t) denote the parameter and signal amplitude under H0, whereas the probability of error Pe given by Pe = p(H0)P f a + p(H1)Pm, in which P f a and Pm denote the probability of false alarm and the probability of miss, respectively, and where p(H0), p(H1) denote the prior probability of the two hypotheses. Without loss of generality, we assume that p(H0) = p(H1) = 1/2. Thus, setting uc = u 1 +u 2 2 and the ARL given by δ = u2 -u1, then, the values of û(t), and ŝ(t) chosen according to the minimal probability of error (MPE) criteria are given by [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF]:

û(t) = uc + γ(t)δ, (6)
ŝ(t) = 1 N b H (û(t)) b(uc - δ 2 )s1(t) + b(uc + δ 2 )s2(t) ,( 7
)
where

γ(t) = |s 2 (t)| 2 -|s 1 (t)| 2 2(|s 2 (t)| 2 +|s 1 (t)| 2 +2ℜ{s * 1 (t)s 2 (t)}) . From the afore- mentioned assumption, it is clear that    H0 : z(t) ∼ CN (µ 0 (t), σ 2 I), where µ 0 (t) = b(û(t))ŝ(t), H1 : z(t) ∼ CN (µ 1 (t), σ 2 I) where µ 1 (t) = 2 k=1 b(u k )s k (t). (8)

First-order Taylor expansions

Assume that the polarization parameters are all equal 1 , i.e. ρ 1 = ρ 2 = ρ 0 and ϕ 1 = ϕ 2 = ϕ 0 . The first-order Taylor expansions of the vectors of interests are considered as follows:

b(u1) = b(uc) - δ 2 ḃ(uc) at (u1 = uc -δ 2 , ρ 0 , ϕ 0 ), b(u2) = b(uc) + δ 2 ḃ(uc) at (u2 = uc + δ 2 , ρ 0 , ϕ 0 ), b(û(t)) = b(uc) + γ(t)δ ḃ(uc) at (u ′ (t) = uc + γ(t)δ, ρ 0 , ϕ 0 )
in which the first-order derivative w.r.t. uc of vector b(uc) = p(ρ 0 , ϕ 0 ) ⊗ a(uc) is defined as ḃ(uc) = p(ρ 0 , ϕ 0 ) ⊗ ȧ(uc) where ȧ(uc) = jdiag{d}a(uc). So, the optimal value (7) becomes

ŝ(t) ∼ = p(t) + δ N κcm(t) (9)
in which

p(t) = s1(t) + s2(t), κc = b H (uc) ḃ(uc), m = [m(1) . . . m(T )] T = V T s where s = [s1(1) s2(1) . . . s1(T ) s2(T )] T and V = Bdiag{v(1), . . . , v(T )} with v(t) = [γ(t) + 1 2 γ(t) - 1 2 ] T .
According to the previous expression, we can see that the optimal source ŝ(t) is approximated by a linear combination of the sources s1(t) and s2(t). Consequently, using the above expression the first-order Taylor expansion of the mean under H0 and H1 can be rewritten as

µ 0 (t) = b(û(t))ŝ(t) ∼ = δν0(t), (10)
µ 1 (t) = 2 k=1 b(u k )s k (t) ∼ = δν1(t). (11)
where

ν0(t) = p(t)γ(t) ḃ(uc) + κcm(t) N b(uc) (12)
ν1(t) = q(t) 2 ḃ(uc). ( 13
)
in which q(t) = s2(t) -s1(t). So, the linearized hypothesis test is given by

H0 : z ∼ = δν0 + n, H1 : z ∼ = δν1 + n (14)
where

z = [z(1) T . . . z(T ) T ] T , n = [n(1) T . . . n(T ) T ] T , ν0 = ν T 0 (1) . . . ν T 0 (T ) T and ν1 = ν T 1 (1) . . . ν T 1 (T ) T .

ARL BASED ON INFORMATION THEORY

Stein's lemma

By maximizing the probability of detection (i.e. P d ≈ 1) for P f a ≤ ǫ with ǫ goes to zero slowly, the best error exponent resulting from using Neyman-Pearson test is given by the Stein's lemma [START_REF] Cover | Elements of information theory[END_REF][START_REF] Jain | Information-theoretic bounds on target recognition performance based on degraded image data[END_REF] as follows: lim

T →∞ ln P f a = -D(p(z|H1) p(z|H0)), (15)
where D(p(z|H1) p(y|H0)) denotes the relative entropy. Let Ω be the observation space. The relative entropy between two Gaussian distributions with parameterized means is given by:

D(p(z|H1) p(z|H0)) = Ω p(z|H1) ln p(z|H1) p(z|H0) dz = 1 σ 2 T t=1 µ 0 (t) -µ 1 (t) 2 .

Geometrical expression of the relative entropy

Using relations [START_REF] Shahram | Imaging below the diffraction limit: A statistical analysis[END_REF] and [START_REF] Cover | Elements of information theory[END_REF], we can link the relative entropy and the ARL according to

D(p(y|H1) p(y|H0)) ∼ = δ 2 σ 2 T t=1 ν0(t) -ν1(t) 2 = δ 2 ||m|| 2 σ 2 κc N b(uc) -ḃ(uc) 2 = δ 2 ||m|| 2 ||d|| 2 σ 2 cos 2 (Θ) using ||p(ρ, ϕ)|| 2 = 1, ||a(uc)|| 2 = N , || ȧ(uc)|| 2 = ||d|| 2 and
Θ is the largest canonical angle between vectors a(uc) and ȧ(uc).

The important point is that the relative entropy can be approximated by a quadratic (in δ) expression. In addition, the relative entropy is a function of the source waveforms, of the array distribution, of the noise variance and of the a useful geometrical quantity which is the "angle" between the steering vector and its first-order derivative. It is interesting to note that the minimal value of 1/ cos 2 (Θ) which is reached for collinear vectors (i.e. Θ = 0) is not relevant since by construction the steering vector and its first-order derivative cannot be collinear. Another geometrical interpretation is: the more orthogonal the two vectors, the smaller the relative entropy. This means that it could be more and more difficult to discriminate the two hypothesis. According to the expression of the relative entropy, we can see that to ensure a "good" discrimination of the two hypothesis, we must have a large ARL or/and a large array distribution and/or a small noise variance.

ARL based on the Stein's lemma

Thus, from the above expressions and using the fact that for optimal P d (close to one), P f a ≈ 2Pe = 2(1η), the ARL takes a closedform expression according to

δ ∼ = -σ log(2) + log(1 -η) µ ||d|| cos(Θ) (16)
where

η > 1/2, µ = √ s H VV T s in which VV T = Bdiag{G(1) . . . G(T )}
where

G(t) = γ(t) + 1 2 2 γ 2 (t) + 1 4 γ 2 (t) + 1 4 γ(t) -1 2 2 . (17)
It is interesting to note that the ARL is affected by the waveform design, (cf. quantity µ) but not from the polarization state in case of unit norm of the polarization vector.

DETECTION THEORY APPROACH

In this Section, we derive the ARL using the detection theory approach, particularly, using the well-known Neyman-Pearson (NP) criterion in the Bayesian context, denoted by BNP. It can be noted that the ARL in this context has not been derived in the existing literature. The BNP will minimize the probability of error Pe. Even if the proposed approach is Bayesian as the one presented by Amar and Weiss [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF], our approach is different. Indeed, Amar and Weiss derive the ARL, denoted Theoretical Resolution Limit (TRL), based on the linearization of the error probability. In our method, we choose to linearize directly the observation signal as done by Sharman and Milanfard [START_REF] Shahram | Imaging below the diffraction limit: A statistical analysis[END_REF].

In order to simplify the calculation, we perform the following change of variable formula:

z ′ = z δ -ν0. (18)
Consequently, plugging (18) into ( 14), one obtains

H0 : z ′ ∼ = n ′ H1 : z ′ ∼ = ζ + n ′ (19)
where ζ = ν1 -ν0 and n ′ ∼ CN (0, σ 2 δ 2 I). Consequently, one has

GNP (z ′ ) = p(z ′ |H1) p(z ′ |H0) H 1 ≷ H 0 τ ′ = p(H0) p(H1) , (20)
denoting TNP (z ′ ) = ln (GNP (z ′ )) and τ = ln (τ ′ ), the statistics test can be given by

TNP (z ′ ) = ln p(z ′ |H1) p(z ′ |H0) = δ 2 σ 2 z ′ -ζ 2 -z ′ 2 = δ 2 σ 2 ζ 2 -2ℜ a H z ′ H 1 ≷ H 0 τ . (21)
Since we have assumed that p(H0) = p(H1) = 1/2, one obtains

H0 : TNP (z ′ ) > 0 H1 : TNP (z ′ ) < 0. (22)
Let L(z ′ ) = ℜ ζ H z ′ , one can easily obtain

H0 : L(z ′ ) ∼ N (0, ̺ 2 ) H1 : L(z ′ ) ∼ N ( ζ 2 , ̺ 2 ) (23)
where

̺ 2 = σ 2 ζ 2 2δ 2 . Thus [13], Pe = 1 2 1 -Q -ζ 2 2 ̺ 2 + Q ζ 2 2 ̺ 2 , ( 24
)
in which Q(.) denotes the right-tail function of the probability function for a Gaussian random variable with zero mean and unit vari-

ance. Since Q -ζ 2 2 √ ̺ 2 = 1 -Q ζ 2 2 √ ̺ 2 , thus, one obtains Pe = Q ζ 2 4̺ 2 (25)
Consequently, the ARL based on the BNP criteria is given by

δ ∼ = σ √ 2Q -1 (1 -η) µ d cos(Θ) (26)
where Q -1 (.) is the inverse of the right-tail function of the probability function for a Gaussian random variable with zero mean and unit variance. In this section, we will compare the ARL based on the Information Theory, on the Bayesian Neyman-Pearson (BNP), and on reference [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF] denoted by TRL for Theoretical Resolution Limit. We consider an uniform linear array consisting of N = 10 vector-sensor components with a half-wavelength equidistantly space. The polarization parameters were chosen as ρ 0 = 1 and ϕ 0 = π/3. The success rate is η = 0.99 thus Pe is set to be 0.01, and the probability of false alarm equals P f a = 0.02. We consider a large number of observations T = 100. Since all parameters are herein assumed to be deterministic, the expectation concerning the TRL in [START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF], will not be performed. The signal to noise ratio SN R is given by SN R = 2 k=1 s k 2 / 2T σ 2 . Fig. 1 shows the behavior of the ARL versus the SNR. One observes that the ARL based on the three methods are very close. It is interesting to note that the ARL is affected by the waveform design but not from the polarization state in case of unit norm of the polarization vector.

CONCLUSION

We have introduced, in this paper, a new approach based on the information theory to obtain the ARL for two closely-spaced polarized sources using a vector-sensor array. The key point is the fact that the relative entropy, in the Stein's lemma, can be approximated by a quadratic function in the ARL. So, it is possible to derive the ARL following this methodology. In addition, a geometrical expression of the ARL is provided. Finally, we compare our approach to the Bayesian Neyman-Pearson test (also derived in this paper) and to the Theoretical Resolution limit (TRL) proposed by Amar and Weiss. In particular, we show that for all these methods, the ARL is a function of the (unit) norm of the polarization vector but not of the specific values of the polarization parameters.

Fig. 1 .

 1 Fig. 1. ARL vs. SNR. The considered ARL are based on the Information Theory, on the Bayesian Neyman-Pearson (BNP), and on reference [8] denoted by TRL for Theoretical Resolution Limit.

Note that this situation is the worst case in the resolution point of view.