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INTRODUCTION

achievement of the objective of halving extreme poverty under the first result implies a rate of growth which is five time larger than the one corresponding to the elasticity calculated by [START_REF] Bhalla | Poor results and poorer policy: A comparative ananlysis of estimates of global inequality and poverty[END_REF] 1 .

As a reaction to this omnipresence of growth, some recent studies, like [START_REF] Gujarati | The Growth Elasticity of Poverty[END_REF], [START_REF] Bourguignon | The Growth Elasticity of Poverty Reduction: Explaining Heterogeneity across Countries and Time Periods[END_REF] or [START_REF] Ravallion | Inequality is Bad for the Poor[END_REF], have emphasized the fundamental role of distribution features in the determination of the poverty rates. Their main message is that growth reduces poverty more efficiently in less inegalitarian countries. [START_REF] Lopez | A Normal Relationship? Poverty, Growth and Inequality[END_REF] also explain that the contribution of inequality reduction to poverty alleviation is higher in richer countries. So, redistribution policies and distributional consequences of growth should not be ignored and the diminution of inequalities should be considered like growth as an intermediate objective of poverty alleviation policies.

However, even if everyone agrees with the necessity of considering the distribution issue for poverty analysis, there is no consensus on the relative contribution of growth and inequality reduction to poverty alleviation. In the case of factors which contribute simultaneously to growth and inequality contraction, it may be of little interest to look for this relative contribution. It is obviously not the situation of all the determinants of growth.

When some factors work in opposite direction -trade openness and financial development are frequently accused of contributing to development at the expense of widening inequalities -it is crucial to know more about the trade-off that are facing decision-makers. Of course, [START_REF] Dollar | Growth is good for the poor[END_REF] results suggest that the growth process is distributionneutral, but, as emphasized by [START_REF] Kanbur | Why is inequality back on the agenda?[END_REF], if the combination of different tools is distribution-neutral on average, it may not be the case for each tool considered independently. Looking for the elasticities of poverty is therefore necessary to build efficient policies to fight poverty.

A direct estimation of these elasticities for different values of mean income and different degree of inequality can easily be achieved under the assumption that the observed distributions can be described by a known statistical distribution. In most studies [START_REF] Quah | Some Simple arithmetic on How Income Inequality and Economic Growth Matter[END_REF][START_REF] Bourguignon | The Growth Elasticity of Poverty Reduction: Explaining Heterogeneity across Countries and Time Periods[END_REF][START_REF] Epaulard | Macroeconomic Performance and Poverty Reduction[END_REF][START_REF] Kalwij | A Decomposition of Poverty Trends across Regions : The Role of Variation in the Income and Inequality Elasticities of Poverty[END_REF][START_REF] Lopez | A Normal Relationship? Poverty, Growth and Inequality[END_REF] the lognormal distribution is used. This is a curious choice since these authors choose to set aside all the XX th century debates on the statistical size distributions of income. Since late XIX th century and the pioneering works of Pareto, research has been extremely active to retrieve the functional form which fits best the observed distributions. Practical considerations and considerable influence of the study of [START_REF] Aitchison | The Lognormal Distribution[END_REF] may still explain the current popularity of the lognormal distribution, but cannot justify its systematic use in empirical studies. Many authors have pointed out its empirical weaknesses and have suggested alternative functional forms 2 like [START_REF] Maddala | A Function for Size Distribution of Incomes[END_REF], [START_REF] Dagum | A new Model of Personal Income Distribution: Specification and Estimation[END_REF] or the general-1 According to [START_REF] Besley | Halving global poverty[END_REF], halving extreme poverty requires that the developing world grows at an annual rate of 3.8% between 1990-2015. With the elasticities suggested by [START_REF] Collier | Can the world cut poverty in half? how policy reform and effective aid can meet international development goals[END_REF] and [START_REF] Bhalla | Poor results and poorer policy: A comparative ananlysis of estimates of global inequality and poverty[END_REF], the needed growth rates is only 1.4% and 0.8% respectively.

2 A quite comprehensive survey is [START_REF] Kleiber | Statistical Size Distributions in Economics and Actuarial Sciences[END_REF].

ized beta 2 distribution [START_REF] Mcdonald | Some Generalized Functions for the Size Distribution of Income[END_REF]. For instance, [START_REF] Bandourian | A Comparison of Parametric Models of Income Distribution across Countries and over Time[END_REF] showed that the lognormal distribution was dramatically outperformed by many alternative functional forms for a relatively large sample of developed countries, even in the set of two parameters distributions. If the lognormal distribution is such a poor approximation of income distributions, elasticities obtained through the lognormality hypothesis are questionable. Of course, this is also the case for all policy recommendations based on this assumption.

In the present paper, we intend to shed light on the consequences of the use of a potentially inadequate distributional hypothesis using alternative distributions which are supposed to fit better observed distributions. Using a data set of 1,842 income distributions for 142 developed and developing countries, we conclude that the quality of our predictions can be significantly improved with more flexible functional forms. Moreover, we find that estimated elasticities under the lognormal assumption generally overestimate "real" elasticities and bias under certain conditions the estimated trade-off between pure growth and redistribution strategies for poverty alleviation in favor of the growth objective.

The paper is organized as follow. The following section introduces the methodology used for the estimation of the desired elasticities. Data and raw results are presented in section 3.

Section 4 is concerned with the criterion used for the choice of an adequate functional form for income distributions and section 5 deals with the drawbacks of the lognormal hypothesis. Section 6 concludes.

Methodology

Calculation of the elasticities of poverty

In the present paper, we focus on absolute poverty. Any absolute poverty measure is a nonlinear combination of a poverty line, the mean income and a set of inequality parameters which fully describes the Lorenz curve L(p). Our preferred measures of poverty are the widely used [START_REF] Foster | A Class of Decomposable Poverty Measures[END_REF] measures P θ :

P θ = z 0 z -y z θ f (y)d y, ( 1
)
where y is income, z the poverty line, f is the income density function and θ the parameter of inequality aversion. For θ = {0, 1, 2}, P θ is respectively the headcount, poverty gap and severity of poverty index. Under the hypothesis that incomes follow a known distribution, f gets a functional form and the set of inequality parameters can be reduced to a few ones.

Our choice for this "analytical" approach is justified in [START_REF] Bresson | How should we estimate the elasticities of poverty?[END_REF]. This approach allows to estimate the required elasticities individually for each observation with few information, to separate perfectly the growth and redistribution effects, and to compute inequality elasticities of poverty that can be compared in cross-section analysis. For the present paper, we are working with the following distributions: Pareto, lognormal, gamma, Weibull, Fisk, Singh-Maddala, Dagum and beta 23 .

To get the parameters of these distributions, we can use some method of moments. With the help of the per-capita income and the Gini index, we can easily obtain the parameters of two parameters distributions like the Pareto, the lognormal, the gamma, the Weibull or the Fisk distributions. Several reasons led us to forsake this approach. The first one is that one needs more information about inequality for more than two parameters distributions.

For large datasets, this supplementary information is usually given by points of the Lorenz curve, but the resolution of the resulting systems of nonlinear equations is generally cumbersome. Second, Gini coefficients are systematically truncated in the available datasets. As we used the points of the Lorenz curve to assess the quality of the fit (cf sec. 4), it appeared that this truncation most of the time increases the size of errors in a significant manner4 .

To avoid these shortcomings, we choose to estimate the parameters of our different distributions uniquely from the available points of the Lorenz curve. The Lorenz curves corresponding to each distribution are presented in table 1.

(p) = 1 -(1 -p) 1-1 α y 0 = µ(α-1) α Lognormal L(p) = Φ Φ -1 (p) -σ ȳ = ln µ -σ 2 2 Gamma L(p) = G G -1 (p, c, γ), c, γ + 1 ρ = µ γ Weibull L(p) = G G W -1 (p, c, β), c, β, 1 + 1 β ρ = µ Γ 1+ 1 β Fisk L(p) = B 1 p, 1 + 1 τ , τ-1 τ κ = µ Γ 1+ 1 τ Γ 1-1 τ Singh-Maddala L(p) = B 1 1 -(1 -p) 1 λ , 1 + 1 τ , λ -1 τ κ = µΓ(λ) Γ 1+ 1 τ Γ λ-1 τ Dagum L(p) = B 1 p 1 θ , θ + 1 τ , 1 -1 τ κ = µΓ(θ) Γ θ+ 1 τ Γ 1-1 τ Beta 2 L(p) = B G2 B 2 -1 (p, c, λ, θ), c, 1, λ + 1, θ -1 κ = µΓ(θ)Γ(λ) Γ(θ+1)Γ(λ-1)
Note: Φ stands for the c.d.f. of the standard normal distribution, c for any constant term, G for the c.d.f. of the Gamma distribution, G G for the c.d.f. of the generalized gamma distribution, W for the c.d.f. of the Weibull distribution, B 1 for the c.d.f. of the Beta distribution of the first kind, B 2 for the c.d.f. of the Beta distribution of the second kind, B G2 for the c.d.f. of the generalized Beta distribution of the second kind. More details on the last distributions in [START_REF] Kleiber | Statistical Size Distributions in Economics and Actuarial Sciences[END_REF].

For the derivation of income and inequality elasticities, we follow [START_REF] Kakwani | Poverty and Economic Growth with Application to Côte d'Ivoire[END_REF]. With the headcount index, the growth elasticity of poverty, η µ is simply:

η 0,µ = - z f (z) P 0 , (2)
where µ stands for mean income. For θ = 0, the elasticity is:

η θ,µ = - θ(P θ-1 -P θ ) P θ . ( 3
)
For the estimation of the inequality elasticity of poverty, we have to deal with the problem that income distributions can change in various ways. Under a strong hypothesis of a two-parameter distribution 5 any variation of an inequality index leads to a unique Lorenz curve. So is it for the inequality elasticities 6 . However, these estimated elasticities cannot be compared through different statistical distributions since they all imply different transformations of the Lorenz curve. For more flexible functional forms, we have to chose how the Lorenz curve should move to get a unique value of the desired elasticity corresponding to given initial conditions. [START_REF] Kakwani | Poverty and Economic Growth with Application to Côte d'Ivoire[END_REF] suggests the following shift of the Lorenz curve:

L * (p) = L(p) -ε p -L(p) , (4)
where ε indicate a proportional change in the Gini coefficient 7 . Such transformation of the Lorenz curve implies Lorenz dominance. So, for negative (positive) value of ε, the situation of the poor never worsen (improve). From equation (4), [START_REF] Kakwani | Poverty and Economic Growth with Application to Côte d'Ivoire[END_REF] proposed the following Gini elasticities:

η 0,G = (µ -z) f (z) P 0 , ( 5
)
η θ,G = θ + µ -z z P θ-1 P θ ∀θ = 0. ( 6
)
5 This is a strong hypothesis since we simultaneously assume that :

• the income distribution that is considered can be described by the chosen statistical distribution;

• the income distribution changes in such a way that the final distribution can also be described by the same kind of statistical distribution.

6 For example, the "natural" Gini elasticity of the headcount index under a strong lognormality assumption is:

η * 0,G = λ   log z µ σ + σ 2     σ 2 - log z µ σ   G σ 2ϕ σ 2 ,
were λ and ϕ represent the hazard rate and density function of the standard normal distribution.

7 It can easily be shown that ε can also be interpreted as the same proportional increase of all the standardized moments of the Lorenz curve defined by [START_REF] Aaberge | Characterizations of lorenz curves and income distributions[END_REF] as:

D t = (t + 1) 1 0 p t -1 p -L(p) d p.
The transformation presented in equation ( 4) is also of great interest because it gives the possibility to compare the inequality elasticities obtained through different distributional assumptions. The interest is not only practical, since the strict respect of a particular distribution for poverty analysis is not neutral in terms of the relative importance of growth and inequality elasticities, and so may reflect some political preferences. With a common strategy for the evolution of inequality and a weak distributional hypothesis -we just assume that the current distribution can be described by the chosen statistical distribution -, estimated elasticities do not depend on the specific distributional changes corresponding to each distribution. On the figure 1 one can observe the differences between several twoparameter distributions which are used in the present study. In each quadrant, the solid line represents the Lorenz curve corresponding to a Gini coefficient equal to 0.55. The dashed and the dotted lines corresponds to the Lorenz curves respectively obtained through the same statistical distribution and through Kakwani's transformation for a 30% decrease of the Gini index. In all cases, we can observe that the Lorenz curve resulting from Kakwani's transformation are more skewed toward the upper point of the Lorenz curve than the "natural" curve corresponding to the same value of the Gini coefficient. This means that the poorest benefit more from the fall of income inequality with Kakwani's transformation. Since the Foster et al. (FGT) measures are directly linked to the slope and curvature of the Lorenz curve, we should generally obtain greater inequality elasticities of poverty than those that are derived from a transformation which preserves the type of the distribution. For the three-parameter distributions that are used in the study, namely the beta of the second kind, the [START_REF] Maddala | A Function for Size Distribution of Incomes[END_REF] -also known as the Burr XII distribution-and the [START_REF] Dagum | A new Model of Personal Income Distribution: Specification and Estimation[END_REF] -also called the Burr III distribution -, elasticities are not unique. So for a same distribution, "natural" elasticities can be either larger or lower than those corresponding to the Kakwani's transformation. However, since the poor are particularly sensitive to this transformation of the Lorenz curve, we can reasonably consider that our estimated Gini elasticities will be quite high.

The other interesting feature of [START_REF] Kakwani | Poverty and Economic Growth with Application to Côte d'Ivoire[END_REF] formula is that the respective importance of growth and inequality elasticities is easily predicted, especially for the headcount index. We note:

η 0,µ η 0,G = z z -µ (7) η θ,µ η θ,G = z(P θ-1 -P θ ) z(P θ-1 -P θ ) -µP θ-1 ∀θ = 0 (8)
Equation ( 7) is particularly interesting since we can see that, for the headcount index, the ratio of the growth elasticity to the Gini elasticity obtained through Kakwani's transformation does not depend on the income distribution. So, it will be the same, whatever distributional assumption is made. As it only depends on per capita income, we can already know that growth policies8 will be more efficient in terms of poverty reduction that redistributive policies when the ratio of the mean income to the poverty line is low. On the contrary, redistribution is the only effective tool for rich countries.

When θ = 0, distribution matters. However, we can notice that the ratio is always negative. It can be easily shown that its absolute value decreases with mean income. So redistributive policies becomes more and more attractive as per capita income increases.

Some alternative functional forms for the Lorenz curve

In addition to known distributions, we tried to use some ad hoc functional forms for the Lorenz curve. Characterizing a distribution through the direct estimation of the Lorenz curve has been first used by [START_REF] Kakwani | On the Estimation of Lorenz Curves from Grouped Observations[END_REF] and has known further important developments. Most of the time, these functional forms are used for descriptive purposes, but [START_REF] Datt | Growth and Redistribution Components of Changes in Poverty Measures: A Decomposition with Applications to Brazil and India in the 1980s[END_REF] suggested that they could be used to estimate elasticities of poverty. These Lorenz curves can be seen as ad-hoc since they are generally not theoretical grounded -the only exception may be [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF]. However, they generally fit pretty well the data and their estimation is rather easy. Nevertheless, the use of ad hoc Lorenz curves raises some problems. First, the underlying distribution function may not be defined for the value of the poverty line9 . Second, the corresponding cumulative distribution functions (c.d.f.) have sometimes no closed form. To calculate the value of the FGT measures and their elasticities, we have to use the following properties of the Lorenz curve10 :

∂L(p) ∂p p=P 0 = z µ , ( 9
)
∂ 2 L(p) ∂p 2 p=P 0 = 1 µ f (z) . ( 10
)
Despite the attractiveness of these functional forms, [START_REF] Datt | Growth and Redistribution Components of Changes in Poverty Measures: A Decomposition with Applications to Brazil and India in the 1980s[END_REF] is the only study in which ad hoc Lorenz curve are used. In the present paper, we include the functional forms11 described by [START_REF] Kakwani | On the Estimation of Lorenz Curves from Grouped Observations[END_REF], [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF], [START_REF] Gaffney | Functional Forms for Estimating the Lorenz Curve[END_REF], [START_REF] Kakwani | On a class of poverty measures[END_REF], [START_REF] Arnold | Elliptical Lorenz Curves[END_REF]Villaseñor (1989), Fernandez, Garcia, Ladoux, Martin and[START_REF] Fernandez | A New Functional Form for Approximating the Lorenz Curve[END_REF] and [START_REF] Chotikapanich | A Comparative of Alternative Functional Form for the Lorenz Curve[END_REF].

All the distributions and Lorenz curves used in the present paper are described in table 2.

Equations for the computation of the poverty headcount for all the used distributions are presented in table 3.

L(p) = p r e -s(1-p) Chotikapanich (1993) L(p) = e kp -1 e k -1 Gaffney et al. (1980) L(p) = 1 -(1 -p) φ 1 ζ Fernandez et al. (1991) L(p) = p ϑ 1 -(1 -p) φ
Maddala and Singh ( 1977)

L(p) = -bd p + (1 -b + bd )p a + b 1 -(1 -p) d Kakwani (1980) L(p) = p -ξp ν (1 -p) υ
Arnold and Villaseñor (1989

) L(p) = f p 2 -L(p) +g L(p)(p-1)+q p-L(p) 1-L(p)
Note: Φ stands for the c.d.f. of the standard normal distribution, G for the c.d.f. of the Gamma distribution, G G for the c.d.f. of the generalized gamma distribution, W for the c.d.f. of the Weibull distribution, B 1 for the c.d.f. of the Beta distribution of the first kind, B 2 for the c.d.f. of the Beta distribution of the second kind, B G2 for the c.d.f. of the generalized Beta distribution of the second kind. More details on the last distributions in [START_REF] Kleiber | Statistical Size Distributions in Economics and Actuarial Sciences[END_REF].

Data and results

Income distribution data are from the UNU-WIDER World Income Inequality Database (version 2.0a., June 2005). Dropping observations when quality and reference population were not satisfying 12 , we get a sample of 1,842 distributions for 142 developed and developing countries from 1950 to 2002. For each distribution we can make use of 6 to 13 points 13 of the Lorenz curve to estimate the parameters of the different Lorenz curves. Most functional forms imply non-linear least squares estimations, but estimators are convergent. To compute our poverty measures and scale parameters, we use PPP per capita income from Penn World Table 6.1. The main characteristics of the database are summarized in tables 11 and 12. For the present exercise, we exclusively work with the traditional 2$PPP poverty line 14, 15 .

Mean elasticities for the various poverty measures and distributions are reported on table 4. Whatever poverty measure we choose, we can observe great differences between most distributions. In the case of the beta 2 distribution, the average growth elasticity of the headcount index is approximately -5.5 but -1.27 with the Weibull distribution. Bootstraped 16

standard errors shows that differences are often significant at the 5% level.

However, one should be careful in interpreting these estimated elasticities. In table 4, we can observe some erroneous values. In particular, we obtain some positive values for the growth elasticities although these are theoretically always negative. These irregular values are due to parameters which do not satisfy the validity conditions of the Lorenz curve (i.e. L(0) = 0, L(1) = 1 and

∂ 2 L(p) ∂p 2
0) or intervals of definitions which do not include the poverty line. The percentage of valid estimations are reported in table 4. It appears that the Pareto distribution and the ad hoc functional forms for the Lorenz curve cannot be employed each one individually to analyze poverty for the whole sample. As the Pareto distributions and [START_REF] Kakwani | On the Estimation of Lorenz Curves from Grouped Observations[END_REF], [START_REF] Kakwani | On a class of poverty measures[END_REF], [START_REF] Arnold | Elliptical Lorenz Curves[END_REF] and [START_REF] Chotikapanich | A Comparative of Alternative Functional Form for the Lorenz Curve[END_REF] Lorenz curves can only be used on a small part of the sample, we temporally exclude them from the set of tools used for the estimation of the elasticities.

With the ramaining functional forms, we can concentrate on a subsample which includes 82.5% of initial observations. For this subsample, the mean income is slightly lower but the difference is not significatively different. The average value of the Gini index is quite equal.

12 In particular, we removed many observations related to urban or rural populations. 13 We add the (0,0) and (1,1) points since some functional forms for the Lorenz curve do not necessarily respect the conditions L(0) = 0 and L(1) = 1. In our sample, the average number of observations is 10. 14 For international comparisons, the 1$PPP poverty line is also widely used. We prefer using the 2$PPP line because it increases the ratio of the poverty line to mean income. Since our sample includes high income countries, it seems more reasonable to adopt the most meaningful line. We also have to mention that a lower poverty line would increase the number of invalid estimations for the ad hoc Lorenz curves since most of them are not defined ∀z ∈ R + . A last reason is that the behavior of most distributions greatly varies at the tails. Our results would presumably be even more heterogeneous with the 1$PPP poverty line. 15 Strictly speaking the exact value is 2.16$ in 1996 PPPs. The poverty line defined for the Millennium Development Goals is fixed for 1993 PPPs, but Penn World Tables 6.1 are based on 1996 values. 16 For the present study, we use a two stage bootstrap procedure. In the first stage, individual elasticities are estimated on many samples with replacement of the points of the Lorenz curve. Then, the required statistics are computed on samples with replacement of the available distributions.

Pareto P 0 = 1 -z y 0 -α Lognormal P 0 = Φ z-µ σ Gamma P 0 = G(z, ρ, γ) Weibull P 0 = 1 -e -z ρ β Fisk P 0 = 1 + z κ -τ -1 Singh-Maddala P 0 = 1 -1 + z κ τ λ Dagum P 0 = 1 + z κ -τ -θ Beta 2 P 0 = B 2 (z, κ, λ, θ)
Kakwani and Podder (1973) (sP 0 + r )P 0 r e -s(1-P 0 ) = z µ Chotikapanich (1993)

P 0 = 1 k log z e k -1 kµ Gaffney et al. (1980) φ ζ 1 -(1 -P 0 ) φ 1 ζ -1 (1 -P 0 ) φ-1 = z µ Fernandez et al. (1991) φP 0 ϑ (1 -P 0 ) φ-1 + ϑP 0 ϑ-1 1 -(1 -P 0 ) φ = z µ Maddala and Singh (1977) -bd + a(1 -b + bd )P 0 a-1 + bd (1 -P 0 ) d-1 = z µ Kakwani (1980) 1 -ξP 0 ν (1 -P 0 ) υ ν P 0 -υ 1-P 0 = z µ
Arnold and Villaseñor (1989

) P 0 = -1 2m n + r g + 2 z µ g + 2 z µ 2 -m -1 w = -f -g -q -1 m = g 2 -4 f n = 2 f w -4q r = n 2 -4mw 2
Note: Φ stands for the c.d.f. of the standard normal distribution, G for the c.d.f. of the Gamma distribution, B 2 for the c.d.f. of the Beta distribution of the second kind. Mean and median values of the elasticities are reported in table 5. Differences between mean elasticities still remain important and frequently significant at the traditional level. In the case of the growth elasticity of the poverty headcount, values are ranged from -4.09 for the lognormal distribution to -1.24 for the Weibull distribution.

However, as noted earlier, distributions generally exhibit different behaviors on the tails. As the absolute values of the elasticities of poverty increase rapidly with mean income, the differences stated in table 4 may only result from extreme values of the calculated elasticities.

To "control" for these extreme values, we reported the median for each elasticity, functional form and poverty measure in table 6. Values are less heterogeneous, but we still notice significant differences.

In both tables 5 and 6, it appears that the lognormal distribution always provide the largest absolute mean and median values of both growth and Gini elasticities. On the contrary, lowest absolute values are provided by the Weibull distribution. However, even if most distributions lead to average growth elasticities close to 2 -a common value in the povertyrelated literature -whatever poverty measure is considered, we cannot tell which value is the good one. Thus, we need to use some criterion to choose between the different functional forms the one that is the most appropriate for poverty analysis.

4 Which distribution should we choose?

As different distributions imply different results for the elasticities of poverty, the question is now to choose the distribution which corresponds best to the empirical distributions of our data set. We assume that we should converge to the true value of the elasticities as the quality of the fit improves. In the statistical size distribution literature, the traditional approach consists in using statistics based on the regression errors. In this way, we calculate the following traditional statistics of goodness-of-fit:

ssr = N i =1 L(p i ) -L(p i ) 2 , ( 11
) sae = N i =1 L(p i ) -L(p i ) , (12)
with L standing for the estimated Lorenz curve.

A problem with these sums of squared (ssr ) and absolute (sae) errors is that all errors are given the same weight. As Datt (1998) notes we are only interested in errors up to the value of the headcount index for the purpose of poverty analysis. In this way, he proposed using the following partial ssr :

p ssr = n i =1 L(p i ) -L(p i ) 2 , ( 13
)
with n corresponding to the first population quantile where p n P0 . However, in the particular case of the headcount index, we are only interested in the quality of the fit in the vicinity of the estimated headcount 17 . Thus, we propose a measure based on squared error with weight decreasing with the distance from the estimated value of the headcount. To make comparisons feasible between each functional form, we normalize the measure by the sum of weights. It also allows us to do comparisons with the traditional ssr statistic. This weighted ssr is:

w ssr = N i =1 L(p i ) -L(p i ) 2 1 -|p i -P0 | 2 N i =1 1 -|p i -P0 | 2 . ( 14
)
However, these criteria just focus on the precision of the estimation, but the use of more flexible functional forms may lead to an improvement of the fit that is not sufficient to compensate for the loss of degrees of freedom. In order to compare non-nested models while 17 Remember that the headcount is defined as the point of the Lorenz curve where the slope is equal to the ratio of the poverty line to mean income, as in equation ( 9). So its elasticities depend only on the form of the Lorenz curve close to the estimated headcount.

penalizing for the addition of new parameters, we can use the Akaike and Schwartz information criteria (cf. [START_REF] Gujarati | The Growth Elasticity of Poverty[END_REF]). These measures are respectively:

ai c = e 2K /N N i =1 L(p i ) -L(p i ) 2 N (15) bi c = N K /N N i =1 L(p i ) -L(p i ) 2 N ( 16
)
where K is the number of estimated parameters.

To ease the comparison of the average value of these different statistics between each functional form, we computed the ratios of mean and median statistics for each distribution to the mean and median of the best-fitted distribution. Results are shown in table 7 for the whole sample (including non-valid estimations) and in table 8 for our restricted sample. It appears that all two-parameter distributions (Pareto, lognormal, gamma, Weibull, Fisk and Chotikapanich 1993) performs very poorly in comparison with three-parameter distributions (beta 2, Singh-Maddala, Dagum, [START_REF] Kakwani | On the Estimation of Lorenz Curves from Grouped Observations[END_REF][START_REF] Gaffney | Functional Forms for Estimating the Lorenz Curve[END_REF][START_REF] Fernandez | A New Functional Form for Approximating the Lorenz Curve[END_REF]) and four-parameter distributions [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF][START_REF] Arnold | Elliptical Lorenz Curves[END_REF][START_REF] Kakwani | Poverty and Economic Growth with Application to Côte d'Ivoire[END_REF]. In particular, we can see that the lognormal is average about 5 to 10 times less precise than [START_REF] Kakwani | On a class of poverty measures[END_REF] and [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF].

This result is not surprising since a single inequality parameter can hardly account for the observed heterogeneity of income distributions. The use of the Akaike and Schwartz information criteria is translated into a fall of the ratios of goodness of fit, but still justifies leaving two-parameter distributions in favor of more flexible functional forms.

Despite the obvious superiority of some functional forms, in particular the [START_REF] Kakwani | On a class of poverty measures[END_REF] and [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF] forms, we cannot reject the other distributions. If a distribution generally poorly fit the data, it does not mean that the fit is systematically poor. To get a more precise picture of the respective performance of each functional form, we ranked the different valid estimations for each observed distribution by their respective value for each statistic. The frequency 18 of first ranking and the median rank of each functional form are reported in table 9. We note that, even if two-parameter distributions are most of the time outperformed by more flexible functional forms, they sometime fit better than more flexible functional forms. In particular, the lognormal distribution is the best choice for 2.7% of the observed income distributions, according to our w ssr statistic. So lognormality is not the rule, but it can be the exception.

Before turning back to elasticities, we have to notice that the observed ranking inside each family of functional forms is rather surprising. An important number of studies have insisted on the merits of the different functional forms used in the present study but comprehensive comparisons of functional forms for income distributions and Lorenz curve are 18 The counts of ranking for each functional form and each criterion are reported in tables 13 to 17. The number of observations in table 15 differs from those of the other tables since per capita income informations are missing for about 250 observations. Since calculation of the w ssr statistic requires this information, the numbers of observations for the ranks are lower. [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF] 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 [START_REF] Bandourian | A Comparison of Parametric Models of Income Distribution across Countries and over Time[END_REF] observed that the Weibull and Dagum were the best-fitting models for the two-and three-parameter distribution family, when opposed to the gamma, lognormal, generalized gamma, beta 1, beta 2 and Singh-Maddala distributions. Our results suggest that Fisk and lognormal distributions are the best two-parameter models and the Singh- Maddala and Gaffney et al. (1980), the best three-parameter models. For ad hoc

Lorenz curves comparisons, [START_REF] Cheong | An Empirical Comparison of Alternative Functional Forms for the Lorenz Curve[END_REF] compared the [START_REF] Kakwani | Efficient Estimation of the Lorenz Curve and Associated Inequality Measures from Grouped Observations[END_REF], [START_REF] Kakwani | On a class of poverty measures[END_REF], [START_REF] Gaffney | Functional Forms for Estimating the Lorenz Curve[END_REF][START_REF] Fernandez | A New Functional Form for Approximating the Lorenz Curve[END_REF] and [START_REF] Chotikapanich | A Comparative of Alternative Functional Form for the Lorenz Curve[END_REF] functional forms on US data from 1977 to 1983 and noticed that [START_REF] Gaffney | Functional Forms for Estimating the Lorenz Curve[END_REF] and [START_REF] Kakwani | On a class of poverty measures[END_REF] were the most powerful models. In the present study, we find that [START_REF] Kakwani | On a class of poverty measures[END_REF] is the functional form which generally fits best to our sample of distributions. However, for poverty analysis, we should prefer [START_REF] Maddala | A Flexible Functional Form for Lorenz Curves[END_REF] and [START_REF] Gaffney | Functional Forms for Estimating the Lorenz Curve[END_REF] forms since definition intervals are larger.

The lognormal case and the most plausible elasticities

In a recent paper, [START_REF] Lopez | A Normal Relationship? Poverty, Growth and Inequality[END_REF], using the Dollar and Kraay ( 2002) database, concluded that lognormality cannot be rejected for the estimation of income distributions. So, even if the lognormal distribution does not fit as well income distributions as other forms do, it may produce reasonable values for the elasticities of poverty. Nevertheless preceding results (cf tables 4 to 6) seem to contradict this assertion. To test this hypothesis, we propose a comparison between estimated lognormal elasticities and the values which seems the most plausible for each observation. To get these values, we simply keep the estimated elasticity corresponding to the best fitting 20 model for each observation. The composition of these series of elasticities is given by the first three columns of table 9. Thus, we get growth and Gini elasticities series for our different poverty measures from the ssr , sae and w ssr statistics. Summary statistics for these series are given in table 10. Since we are only interested in the quality of the fit in the vicinity of the estimated headcount, the mixed-series based on the w ssr criterion will be our preferred series.

It appears that lognormal elasticities are most of the time higher in absolute value than those obtained in the mixed series. On average, the lognormal elasticities are about 1 to 2 percentage points higher in absolute value for the growth elasticities. Differences in Gini elasticities are really striking, in particular for the headcount index. Under the lognormality assumption, the average elasticity is 78.43, twice larger than our preferred value of the elasticity. Such a large difference cannot be attributed to extreme values since the same 19 Most of the time, goodness-of-fit tests of ad hoc Lorenz curves include the lognormal distribution as a benchmark, but never more performing distributions 20 As a robustness test, we also designed series corresponding to estimations that were ranked second according to our different statistics. Result for these second-best series do not differ from those obtained with best-fitting series. phenomenon is observed for median values. However, overestimation of the Gini elasticities tends to shrink as the θ parameter of the FGT measure increases.

Overestimation is a major problem, but we can imagine that lognormal elasticities are highly correlated with "true" elasticities, so that we can find the appropriate values of the desired elasticities through a simple linear correction. Figure 2 clearly show that the correlation between lognormal and the mixed series based on the w ssr criterion is rather low 21 (from 0.3 to 0.5 depending on the elasticity and the goodness-of-fit criterion). The plots also confirm that, on average, the lognormal assumption overestimates, in absolute value, the growth and Gini elasticities of poverty. So, we should be extremely cautious with simulations based on the lognormality assumption 22 .

-15 -10 -5 0 Finally, even if the lognormal distribution is not satisfying for a correct prediction of the magnitude of each elasticity, we may use it to compare the relative effectiveness of pure growth and pure distributional policies to achieve poverty alleviation. In section 2.1 (equation 7), we showed that under assumption (4) the ratio of the growth to Gini elasticities is 21 The same differences are observed with the mixed series based on the ssr and sae statistics. Plots are reported on figures 6 and 7. 22 Such risks are perfectly illustrated in CGE models by [START_REF] Boccanfuso | Poverty, Income Distribution and CGE modeling : Does the Functional Form of Distribution Matter?[END_REF] independent of the distribution23 when the headcount index is considered.This is not the case for higher values of the θ parameter of the FGT class of poverty measures. In order to check whether the lognormal assumption biases policy recommendations in favor of growth or distributional objectives, we can use the ratio of the lognormal elasticities ratio to the one corresponding to our best-fitting estimations. A value greater (lower) than unity for this policy bias ratio indicates that the lognormal assumption bias politics towards growth (inequality reduction). A kernel estimation of the density of this ratio for the P 1 and P 2 measures is reported on figure 3. As the mean and median value of the policy bias ratio are close to unity, it seems that the use of the lognormality assumption does not systematically bias analysis in favor of a particular kind of policy. However, the variance of this ratio is large, in particular for the P 2 measure. Under the lognormal hypothesis, we notice that the relative importance of the growth elasticity can be overestimated or underestimated in excess of 50%. So we should be cautious with the use of the lognormal distribution to estimate the growth-inequality tradeoff.

This large variance may be explained by the presence of high income countries in our sample. As poverty defined with a 2 $ PPP poverty line is essentially a concern for low income countries, it seems important to check if the mean value of our policy bias ratio and its variance vary with the level of development. On figure 4, we can observe the result of a non-parametric 24 estimation of the mean of this ratio conditional to income per capita. The gray area represents the 95% confidence interval of this conditional mean using a bootstrap procedure. We notice that the estimated ratio is significantly different from unity for income greater than $4000. So, when the ratio of the per capita income on the poverty line is greater than 5, the use of the lognormal assumption bias our policy recommendations in favor the growth objective. Surprisingly we also observe an under-estimation of the ratio for values in the viciny of the poverty line. Note: solid and dashed lines respectively correspond to mean and median values. The thick line corresponds to the gaussian kernel estimation of the conditional mean of the political bias ratio. The gray area represents the 95% confidence interval of this conditional mean using a bootstrap procedure (500 replications, see note 16). Finally, to test if the lognormality assumption imply some policy bias depending on the degree of inequality, we also realized a non-parametric estimation of the mean of our policy bias ratio conditional to the inequality degree. Results are reported on figure 5. We can see that the relative effect of growth is over-estimated under the lognormality hypothesis when the Gini coefficient is less than 0.45. Since more than 70% of our sample includes distributions which exhibits Gini index that are less than 0.45, this result cannot be considered as trivial.

Concluding remarks

Throughout the present paper, we intended to answer the following questions. Which distributional hypothesis is suitable for a good estimation of the growth and inequality elasticities of poverty? What are the consequences of the use of a distributional assumption that does not suit to the observed heterogeneity of empirical income distributions? These are important questions since standard values of the elasticities of poverty are often used in applied studies like [START_REF] Collier | Can the world cut poverty in half? how policy reform and effective aid can meet international development goals[END_REF], although uncertainty is great concerning the real val- Note: solid and dashed lines respectively correspond to mean and median values. The thick line corresponds to the gaussian kernel estimation of the conditional mean of the political bias ratio. The gray area represents the 95% confidence interval of this conditional mean using a bootstrap procedure (500 replications, see note 16). For the first question, we conclude that in the absence of very flexible functional forms which parameters can easily be estimated, none of the tested distributions performs systematically better than the others for the description of observed income distributions, and so for poverty analysis. Pragmatism is required and we have to choose for each distribution the distributional assumption that fits best data. However, we can assert that two-parameter distributions should not be used in cross-section studies. More flexible functional forms are more appropriate for dealing with the heterogeneity of observed income distributions.

Our main result is that the use of poor distributional assumptions may induce some biases in the analysis of poverty. We chose to illustrate these biases with a comparison of the elasticities obtained through the popular lognormality assumption with those corresponding to the functional form that fits best each observed distribution. We notice that the use of the lognormal distribution leads to an overestimation of the effects of growth and inequality reduction in terms of poverty alleviation. Consequences are probably worse concerning the estimation of the relative size of these effects for the design of efficient policies aimed at poverty reduction. In particular we showed that resorting to the lognormal hypothesis introduces a significant bias in favor of growth oriented policies for high income or low and moderately unequal countries. Such biases may explain why it seems so difficult to reach the goals of poverty alleviation in many developing countries.

Appendix

FiskFigure 1 :

 1 Figure 1: "Natural" vs Kakwani's transformation of the Lorenz curve for two parameters distributions.

  and horizontal solid and dashed lines respectively correspond to mean and median values.

Figure 2 :

 2 Figure 2: Comparison of the estimated elasticities of poverty between lognormal distribution and wssr series.

  solid and dashed lines respectively correspond to mean and median values.

Figure 3 :

 3 Figure 3: Gaussian kernel density of the policy bias ratio for P 1 and P 2 (lognormal against best-fitting estimations).

Figure 4 :

 4 Figure 4: GDP per capita versus policy bias ratio for P 1 and P 2 (lognormal against mixed-series wssr): non-parametric regression.

Figure 5 :

 5 Figure 5: Gini index versus policy bias ratio for P 1 and P 2 (lognormal against mixed-series wssr): non-parametric regression.

  and horizontal solid and dashed lines respectively correspond to mean and median values.

Figure 6 :

 6 Figure 6: Comparison of the estimated elasticities of poverty between lognormal distribution and ssr series.

Table 1 : The Lorenz curve of used classical distributions

 1

	Name	Lorenz curve	Scale parameter
	Pareto	L

Table 2 : The different ad hoc functional forms used

 2

	Name	Lorenz curve
	Kakwani and Podder (1973)

Table 3 : Calculation of the headcount index for the different functional forms

 3

	Name	Headcount index (c.d.f.)

Table 4 : Mean value of growth and Gini elasticities of P 0 , P 1 and P 2 : whole sample.

 4

	Distribution	Growth elasticity		Gini elasticity		Valid
		P 0	P 1	P 2	P 0	P 1	P 2	estimations (%)
	Pareto	2	2.57	3.41	-22.92	-17.15	-14.07
		(5.29)	(9.63)	(10.52)	(11.24)	(9.82)	(14.15)
	Lognormal	-4.73	-5.06	-5.3	78.43	90.82	102.83	100
		(0.14)	(0.13)	(0.15)	(2.97)	(3.93)	(3.8)
	Gamma	-1.81	-1.89	-1.94	24.53	34.65	44.7	100
		(0.07)	(0.07)	(0.1)	(1.31)	(1.22)	(2.6)
	Weibull	-1.27	-1.33	-1.36	15.32	25.28	35.21	100
		(0.05)	(0.05)	(0.05)	(0.64)	(0.92)	(1.03)
	Fisk	-2.21	-2.33	-2.41	27.09	37.09	47.04	100
		(0.05)	(0.03)	(0.04)	(0.81)	(1.07)	(1.38)
	Beta 2 Singh-Maddala	-5.5 (0.86) -1.93	-5.76 (0.87) -2.05	-5.96 (26.54) -2.12	100.89 (24.76) 23.18	112.15 (21.1) 33.17	123.2 (13 × 10 2 ) 43.12	100 100
		(0.07)	(0.08)	(0.67)	(1.11)	(1.37)	(9.19)
	Dagum	-1.88	-2	-2.08	22.43	32.42	42.37	100
		(0.17)	(0.17)	(0.19)	(1.80)	(2.06)	(2.13)
	Kakwani and Podder (1973) Arnold and Villaseñor (1989)	-68 × 10 6 (22 × 10 9 ) (61.48) -4.35 -0.91 -0.89	-4.89 (0.53) -0.85	12 × 10 7 (2 × 10 11 ) 1.37	51.77 (39.73) 3.09	82.6 (4.06) 4.87
		(22.24)	(57.75)	(16.5)	(300.73)	(720.72)	(176.07)
	Chotikapanich (1993)	1.04	2.42	3.89	-6.99	-7.87	-9.86
		(5)	(16.11) (15.87)	(19.01)	(73.36)	(83.06)
	Gaffney et al. (1980)	-2.09	-2.09	-2.17	23.73	33.48	43.45
		(0.21)	(0.09)	(0.09)	(4.75)	(1.45)	(1.69)
	Fernandez et al. (1991) Kakwani (1980)	-1.86 (14 × 10 4 ) -0.68	-1.97 (0.09) -1.59	-2.04 (0.2) -1.22	22.27 (7 × 10 4 ) 3.4	32.25 (1.44) 14.47	42.2 (1.75) 17.59
		(0.13)	(2.28)	(0.46)	(0.99)	(17.57)	(3.05)
	Maddala and Singh (1977)	40.3 × 10 4 (11 × 10 6 )	-2.6 (0.48)	-2.61 (0.31)	-41 × 10 5 (14 × 10 7 )	39.01 (4)	40.63 (1.98)

Note: bootstraped standard errors in parentheses.

Table 5 : Mean value of growth and Gini elasticities of P 0 , P 1 and P 2 : valid common sample.

 5

	Distribution	Growth elasticity	Gini elasticity
		P 0	P 1	P 2	P 0	P 1	P 2
	Lognormal	-4.09	-4.42	-4.67	56.12 67.31 78.14
		(0.2) (0.23) (0.25)	(5.74) (6.08) (6.07)
	Gamma	-1.66	-1.75	-1.81	18.33 27.49 36.57
		(0.08) (0.09) (0.09)	(1.55) (2.23) (2.48)
	Weibull	-1.24	-1.3	-1.34	12.52 21.52	30.5
		(0.05) (0.05) (0.06)	(0.87) (1.2) (1.83)
	Fisk	-2.16	-2.29	-2.36	22.58 31.62 40.62
		(0.05) (0.05) (0.05)	(1.42) (1.92) (2.45)
	Beta 2	-3.12	-3.39	-3.58	34.24 44.05 53.66
		(0.25) (0.26) (0.27)	(4.38) (4.65) (4.96)
	Singh-Maddala	-1.85	-1.98	-2.05	18.07 27.12 36.12
		(0.06) (0.06) (0.06)	(1.05) (1.44) (1.81)
	Dagum	-1.75	-1.88	-1.96	16.33 25.37 34.38
		(0.09) (0.09) (0.1)	(1.17) (1.43) (1.76)
	Gaffney et al. (1980)	-2	-2.01	-2.09	18.29 27.09 36.11
		(0.06) (0.07) (0.07)	(1.1) (1.31) (1.9)
	Fernandez et al. (1991)	-1.77	-1.89	-1.96	17	26.03 35.03
		(0.43) (0.07) (0.08)	(0.99) (1.52) (1.55)
	Maddala and Singh (1977) -2.25	-2.46	-2.61	21.5	30.9	40.17
		(0.16) (0.26) (0.29)	(1.69) (2.11) (2.18)

Note: bootstraped standard errors in parentheses. Common sample: 82.5% of initial observations; mean income: 6,963 $PPP; mean Gini: 0.39.

Table 6 : Median value of growth and Gini elasticities of P 0 , P 1 and P 2 : valid common sample.

 6

	Distribution	Growth elasticity	Gini elasticity
		P 0	P 1	P 2	P 0	P 1	P 2
	Lognormal	-2.66	-3.05	-3.34	13.01 20.56 27.66
		(0.2)	(0.2) (0.21)	(1.3) (1.66) (1.97)
	Gamma	-1.43	-1.54	-1.6	6.1	12.47	18.3
		(0.12) (0.13) (0.12)	(0.69) (0.87) (1.04)
	Weibull	-1.26	-1.29	-1.32	5.46	11.3	17.14
		(0.08) (0.08) (0.08)	(0.44) (0.73) (0.97)
	Fisk	-2.28	-2.39	-2.45	10.75 16.54 22.44
		(0.08) (0.09) (0.08)	(0.75) (0.98) (1.26)
	Beta 2	-2.35	-2.58	-2.71	12.14 19.31 26.24
		(0.11) (0.11) (0.12)	(1.28) (1.6)	(1.9)
	Singh-Maddala	-1.86	-1.92	-1.97	9.11	15.14 21.15
		(0.06) (0.07) (0.06)	(0.69)	(1)	(1.37)
	Dagum	-1.65	-1.7	-1.75	8.32	14.31 20.14
		(0.09) (0.09) (0.1)	(0.67) (0.97) (1.21)
	Gaffney et al. (1980)	-1.96	-1.92	-1.97	9.7	15.29 21.19
		(0.07) (0.07) (0.08)	(0.63) (0.97) (1.39)
	Fernandez et al. (1991)	-1.73	-1.78	-1.84	8.5	14.48 20.34
		(0.4) (0.08) (0.08)	(0.69) (0.99) (1.24)
	Maddala and Singh (1977) -2.03	-2.15	-2.21	10.75 18.16 24.22
		(0.21) (0.17) (0.21)	(1.29) (1.55)	(2)
	Note: bootstraped standard errors in parentheses. Common sample: 82.5% of initial obser-
	vations; mean income: 6,963 $PPP; mean Gini: 0.39.

Table 7 : Ratios of goodness-of-fit.

 7

	Distribution	ssr		sae		w ssr	ai c	bi c
		Mean	Median	Mean Median	Mean Median	Mean	Median	Mean	Median
	Pareto	98.82	133.79	14.55	15.34	112.10	161.57	61.75	101.54	60.10	93.20
	Lognormal	8.92	4.51	3.26	2.50	9.57	4.88	5.91	3.74	5.72	3.41
	Gamma	30.17	20.16	6.53	5.24	33.92	21.86	18.71	17.28	18.23	15.57
	Weibull	32.07	31.47	7.43	6.99	35.50	35.26	20.03	26.21	19.50	23.90
	Fisk	9.50	5.03	3.37	2.56	9.10	4.87	6.31	3.88	6.10	3.60
	Beta 2	2.29	1.30	1.51	1.29	2.56	1.36	2.49	1.29	2.31	1.23
	Singh-Maddala	1.88	1.38	1.40	1.30	1.97	1.56	1.63	1.18	1.59	1.13
	Dagum	1.79	1.49	1.49	1.36	1.97	1.75	1.52	1.30	1.49	1.25
	Kakwani and Podder (1973)	89.82	70.88	8.86	8.08	65.35	16.58	64.76	69.51	65.13	64.59
	Arnold and Villaseñor (1989) 22032.78	1.40	12.92	1.28	113.35	1.31	49939.50	1.39	42958.82	1.38
	Chotikapanich (1993)	48.11	34.53	7.90	6.46	53.90	42.35	29.45	28.88	28.73	26.08
	Gaffney et al. (1980)	1.45	1.34	1.33	1.26	1.57	1.48	1.23	1.13	1.21	1.09
	Fernandez et al. (1991).	1.64	1.52	1.47	1.40	1.84	1.87	1.38	1.29	1.35	1.24
	Kakwani (1980)	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
	Maddala and Singh (1977)	1.07	1.05	1.06	1.07	1.06	1.04	1.12	1.05	1.11	1.04

Table 8 : Ratios of goodness-of-fit (restricted valid sample).

 8

	Distribution	ssr		sae		w ssr		ai c		bi c
		Mean Median	Mean Median	Mean Median	Mean Median	Mean Median
	Lognormal	8.49	4.41	3.15	2.37	10.21	5.01	5.74	3.81	5.54	3.51
	Gamma	36.59	20.31	6.69	4.93	40.49	20.54	23.31	17.23	22.65	15.96
	Weibull	38.94	30.63	7.61	6.55	43.40	34.05	25.03	26.35	24.29	24.43
	Fisk	7.69	4.58	3.05	2.33	7.68	4.55	5.32	3.55	5.12	3.28
	Beta 2	1.77	1.21	1.36	1.17	2.22	1.28	1.83	1.20	1.72	1.14
	Singh-Maddala	1.36	1.29	1.26	1.19	1.50	1.46	1.12	1.10	1.10	1.06
	Dagum	1.60	1.38	1.37	1.25	1.85	1.63	1.30	1.17	1.28	1.14
	Gaffney et al. (1980)	1.35	1.23	1.23	1.15	1.49	1.38	1.10	1.06	1.08	1.02
	Fernandez et al. (1991)	1.52	1.40	1.37	1.27	1.72	1.72	1.22	1.21	1.20	1.17

Table 9 : Goodness of fit: frequency of first ranking and median rank.

 9 Moreover, they rarely mix classical distributions and ad hoc Lorenz curves 19 . Using 82 distribution data sets at various years for 23 developed and middle-income countries,

	Distribution	Frequency of first ranking (%)		Median rank
		ssr	se a w ssr ai c	bi c	ssr se a w ssr ai c bi c
	Pareto	0.0	0.0	0.1	0.1	0.1	12	13	12	12
	Lognormal	1.2	2.4	2.7	9.3 10.4	8	8	8	8
	Gamma	0.9	0.7	1.0	5.0	5.3	9	9	9	9
	Weibull	0.2	0.3	0.3	1.0	1.2	10	10	10	10
	Fisk	0.0	1.0	0.3	4.6	5.3	8	8	8	8
	Beta 2	17.8 16.5	20.0 11.9 13.4	3	3	4	4
	Singh-Maddala	8.9	8.5	6.5 13.0 12.6	4	4	5	3
	Dagum	6.9	7.5	5.3	7.9	8.0	5	5	6	5
	Kakwani and Podder (1973)	0.3	0.4	0.8	0.2	0.3	12	12	11	12
	Arnold and Villaseñor (1989)	5.9	6.4	7.8	5.3	5.1	4	3	3	4
	Chotikapanich (1993)	0.3	0.3	0.3	0.4	0.4	12	12	12	12
	Gaffney et al. (1980)	3.5	5.5	5.4 10.4	9.8	4	4	4	4
	Fernandez et al. (1991)	4.2	4.0	3.9	7.8	7.9	6	6	6	6
	Kakwani (1980)	21.0 19.9	20.3 10.6	8.7	1	1	2	2
	Maddala and Singh (1977)	29.0 26.7	25.5 12.5 11.6	2	2	2	4

Table 10 : Summary statistics for the mixed series based on the ssr, sae and wssr statistics.

 10

	Statistics		P 0			P 1			P 2
		ssr	se a	w ssr	ssr	se a	w ssr	ssr	se a	w ssr
	Growth elasticities
	Mean Min 1 st quartile Median 3 rd quartile Max Std deviation	-2.44 -34.11 -2.97 -1.54 -0.89 -0.01 3.65	-2.42 -34.11 -2.93 -1.58 -0.90 -0.01 3.54	-2.90 -55.33 -3.24 -1.74 -0.91 -0.01 3.99	-3.10 -51.03 -3.55 -2.13 -1.26 -0.14 4.95	-3.08 -51.03 -3.47 -2.12 -1.26 -0.14 4.91	-3.49 -55.72 -3.79 -2.22 -1.29 -0.14 5.01	-3.25 -49.35 -3.66 -2.20 -1.26 0.00 11.80	-3.24 -49.35 -3.60 -2.17 -1.26 0.00 11.79	-3.67 -56.09 -4.02 -2.34 -1.30 0.00 11.82
	Gini elasticities
	Mean	33.49	32.73	41.69	46.92	46.10	54.96	56.90	56.24	65.26
	Min 1 st quartile	-0.31 1.30	-0.31 1.30	-0.31 1.30	0.00 4.03	0.00 4.03	0.00 3.99	0.06 6.52	0.06 6.49	0.06 6.52
	Median	6.43	6.78	7.56	18.49	18.51	19.12	24.83	24.99	25.71
	3 rd quartile	37.00	37.45	41.12	58.96	58.76	62.53	74.20	74.20	77.96
	Max	1887.65 1887.65 1887.65	1957.24 1957.24 1957.24	2026.43 2026.43 2026.43
	Std deviation	94.55	90.67	98.76	104.72	101.09	109.29	177.56	175.39	180.84

Table 11 : Distribution of the data in time and region.

 11

	Period	EAP LAC NA MENA SA SSA EEAC WE Total
	1950-54	4	7	0	1	11	0	0	4
	1955-59	13	9	0	2	12	7	0	6
	1960-64	23	29	0	3	15	8	5	21
	1965-69	28	22	8	3	20	19	10	24
	1970-74	35	36	11	2	17	1	9	39
	1975-79	27	28	11	4	11	7	7	29
	1980-84	32	27	10	1	9	7	8	46
	1985-89	38	54	13	5	18	28	30	69
	1990-94	49	70	13	6	6	58	43	74
	1995-99	49	83	7	11	8	38	71 131
	2000-02	11	20	5	3	3	7	12	81
	Total	309 385	78	41 130 180	195 524

EAP: East Asia and Pacific; LAC: Latin America and Caribbean; NA: North America; MENA: Middle East and North Africa; SA: South Asia; SSA: Sub-Saharan Africa; EECA: East Europe and Central Asia; WE: Western Europe.

Table 12 : Descriptive statistics of the dataset.

 12

	Statistic	GDP per capita ($PPP) Gini coefficient
	Mean	7705, 21	0, 39
	Minimum	138, 77	0, 17
	1 st quartile	1667, 04	0, 32
	Median	4673, 94	0, 37
	3 rd quartile	11522, 19	0, 47
	Maximum	48967, 56	0, 79

Table 13 : Goodness-of-fit rank: ssr statistic.

 13

	Distribution	1	2	3	4	5	6	7	8	9	10	11	12	13	14
	Pareto	0	1	0	0	1	4	11	16	26	55	32	83	91	70
	Lognormal	22	85	62	90 239	98 227 278 433 256	41	12	1	0
	Gamma	16	39	35	38	58 254 117 153 302 324 450	58	0	0
	Weibull	3	10	14	26	28	63 277 113 251 486 234 238	86	14
	Fisk	0	1	34 129 123 133 241 442 296 233 128	55	28	1
	Beta 2	328 332 343 149	72 193 236 143	41	6	1	0	0	0
	Singh-Maddala	165 257 352 384 310 243 101	32	0	0	0	0	0	0
	Dagum	127 208 202 285 302 287 262 116	51	4	0	0	0	0
	Kakwani and Podder (1973)	6	4	0	0	0	0	5	2	13	38 165 121 131 134
	Arnold and Villaseñor (1989) 108 123	67	94	73	53	36	21	12	6	1	2	0	0
	Chotikapanich (1993)	5	3	0	4	2	1	2	3	10	33 153 325 225	2
	Gaffney et al. (1980)	65 184 281 372 377 167	43	4	0	0	0	0	0	0
	Fernandez et al. (1991)	77 126 147 155 190 313 260 175	58	1	0	0	0	0
	Kakwani (1980)	387	95	79	48	26	11	4	2	0	0	0	0	0	0
	Maddala and Singh (1977)	535 376 228	70	43	24	22	10	3	0	0	0	0	0

Table 14 : Goodness-of-fit rank: sea statistic.

 14

	Distribution	1	2	3	4	5	6	7	8	9	10	11	12	13 14
	Pareto	0	1	0	0	2	4	11	15	23	49	26	70	96 89
	Lognormal	45	68	42	82 224 116 236 264 428 275	39	24	1	0
	Gamma	12	29	32	36	53 267 135 145 307 332 449	45	2	0
	Weibull	6	9	10	23	33	58 278 118 239 472 209 218 144 27
	Fisk	19	31	63 147 129 144 223 384 280 189 123	72	34	6
	Beta 2	304 326 330 166	71 200 232 172	37	5	1	0	0	0
	Singh-Maddala	156 256 360 350 316 247 120	39	0	0	0	0	0	0
	Dagum	139 222 205 261 313 276 257 113	51	5	2	0	0	0
	Kakwani and Podder (1973)	7	3	1	0	0	1	4	7	32	53 169 102 144 99
	Arnold and Villaseñor (1989) 118 126	85	78	84	42	27	20	8	6	1	1	0	0
	Chotikapanich (1993)	5	3	1	4	1	0	1	5	11	48 186 362 141	0
	Gaffney et al. (1980)	101 172 253 369 354 177	59	8	0	0	0	0	0	0
	Fernandez et al. (1991)	73 168 163 139 172 266 234 202	77	8	0	0	0	0
	Kakwani (1980)	367	94	66	71	30	14	10	0	0	0	0	0	0	0
	Maddala and Singh (1977)	492 336 233 118	62	32	17	18	3	0	0	0	0	0

Table 15 : Goodness-of-fit rank: wssr statistic.

 15

	Distribution	1	2	3	4	5	6	7	8	9	10	11	12	13 14
	Pareto	1	0	3	2	1	1	15	15	32	60	50	80	69 63
	Lognormal	42	82	81	55 108	69 188 219 347 285	73	23	6	0
	Gamma	15	32	16	40	33 110	83 127 258 280 370 184	30	0
	Weibull	4	9	14	13	16	17	92	86 225 386 265 208 182 61
	Fisk	5	17	54	96	70 110 146 368 282 150 136	92	40 12
	Beta 2	316 295 224 120	66 135 195 162	53	12	0	0	0	0
	Singh-Maddala	102 144 245 294 313 268 165	46	1	0	0	0	0	0
	Dagum	83 134 130 153 258 265 282 172	79	16	6	0	0	0
	Kakwani and Podder (1973)	12	7	2	2	5	6	14	46	88 124	85	80	77 68
	Arnold and Villaseñor (1989) 123 116	63	98	69	49	33	20	10	7	6	2	0	0
	Chotikapanich (1993)	5	3	0	4	1	6	0	5	20 110 214 225 158 17
	Gaffney et al. (1980)	86 156 262 369 384 176	54	5	1	0	0	0	0	0
	Fernandez et al. (1991)	61 115 131 135 158 309 269 217	96	11	0	0	0	0
	Kakwani (1980)	321 137	93	62	25	7	4	3	0	0	0	0	0	0
	Maddala and Singh (1977)	402 331 260 135	71	50	38	19	4	1	0	0	0	0

Table 16 : Goodness-of-fit rank: aic statistic.

 16

	Distribution	1	2	3	4	5	6	7	8	9	10	11	12	13	14
	Pareto	1	0	2	1	0	3	12	15	25	55	33	82	96	74
	Lognormal	172	84	57	70 207 125 171 250 407 251	39	11	0	0
	Gamma	92	28	31	34	53 258 102 132 283 323 450	58	0	0
	Weibull	18	26	16	22	25	56 270 113 243 483 233 238	86	15
	Fisk	85	46	83 121 128 121 186 369 266 245 119	47	27	1
	Beta 2	220 291 318 209 100 135 292 187	75	13	4	0	0	0
	Singh-Maddala	239 315 373 296 267 194 117	42	1	0	0	0	0	0
	Dagum	145 265 198 281 252 244 257 139	55	7	1	0	0	0
	Kakwani and Podder (1973)	4	5	0	1	0	0	2	5	11	24 170 131 127 129
	Arnold and Villaseñor (1989)	98	86	51	70	62	72	73	42	28	7	4	2	1	0
	Chotikapanich (1993)	8	0	1	6	1	1	4	1	10	32 152 325 225	2
	Gaffney et al. (1980)	191 271 246 310 309 128	35	3	0	0	0	0	0	0
	Fernandez et al. (1991)	144 121 162 151 163 277 229 173	80	2	0	0	0	0
	Kakwani (1980)	196 134	94	84	72	44	21	6	1	0	0	0	0	0
	Maddala and Singh (1977)	231 172 212 188 205 186	73	33	11	0	0	0	0	0

Table 17 : Goodness-of-fit rank: bic statistic.

 17

	Distribution	1	2	3	4	5	6	7	8	9	10	11	12	13	14
	Pareto	1	0	3	0	0	3	11	16	25	55	33	82	97	74
	Lognormal	191	74	58	69 209 121 169 248 408 249	37	11	0	0
	Gamma	97	28	28	35	50 257 100 132 286 323 450	58	0	0
	Weibull	22	22	16	24	23	57 272 108 245 483 233 238	86	15
	Fisk	98	50	82 128 115 123 186 357 266 243 120	48	27	1
	Beta 2	247 320 318 189	92 127 292 182	60	13	4	0	0	0
	Singh-Maddala	233 326 368 309 257 197 106	47	1	0	0	0	0	0
	Dagum	148 263 204 294 239 229 260 140	59	6	2	0	0	0
	Kakwani and Podder (1973)	6	3	0	1	0	0	4	3	11	26 169 130 126 129
	Arnold and Villaseñor (1989)	94	76	51	68	57	85	63	52	35	7	5	2	1	0
	Chotikapanich (1993)	7	1	1	6	0	2	4	2	9	32 152 325 225	2
	Gaffney et al. (1980)	180 282 258 311 302 121	37	2	0	0	0	0	0	0
	Fernandez et al. (1991)	145 121 169 155 171 266 218 171	81	5	0	0	0	0
	Kakwani (1980)	161 124	95	80 111	46	27	7	1	0	0	0	0	0
	Maddala and Singh (1977)	214 154 193 175 218 210	95	43	9	0	0	0	0	0

We also tried to use the generalized gamma, the beta of the first kind and the generalized beta of the second kind distributions. As the estimators of the non-linear least-squares were not convergent with these functional forms, we gave up using these distributions. For a closer look at the linkages between all these distributions, see[START_REF] Mcdonald | Some Generalized Functions for the Size Distribution of Income[END_REF].

Truncations and rounding are also a matter of concern for points of the Lorenz curve, but the loss of precision is less important.

METHODOLOGY

By growth policies, we mean policies that would lead to an increase of mean income with no distributional change. Of course, this is a pure theoretical view since observed growth always implies some redistribution.

This is also a well-known feature of the Pareto distribution.

More details on the use of ad hoc Lorenz curves for poverty analysis in[START_REF] Datt | Computational Tools for Poverty Measurement and Analysis[END_REF]

We also tried to estimate the parameters of[START_REF] Castillo | An Ordered Family of Lorenz Curves[END_REF] class of Lorenz curve, but estimators were not convergent.

WHICH DISTRIBUTION SHOULD WE CHOOSE?

THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES

Of course, this ratio is not independent of the distributional policy which can be used.

Estimations are realized with a gaussian kernel. Bandwidth is chosen using the cross-validation procedure.

CONCLUDING REMARKS