the linear mixed models with both random and fixed effects have been extensively and actively studied in recent years from both theoretical and applied aspects in the literature . as specific normal linear mixed models ,the fay - herriot model ( fay and herriot , 1979 ) and the nested error regression models ( battese , harter and fuller , 1988 ) have been used in small - area estimation since direct estimates like sample means for small areas have unacceptable estimation errors because of small sample sizes in small areas .the model - based shrinkage methods such as the empirical best linear unbiased predictor ( eblup ) are very useful for providing reliable estimates for small - areas with higher precisions by borrowing data in the surrounding areas .recently , several approaches for small area estimation are proposed and investigated in terms of both parametric and nonparametric aspects .for example , see hall and maiti ( 2006a , b ) , chambers ( 2014 ) , chaudhuri and ghosh ( 2011 ) , jiang and nguyen ( 2012 ) and opsomer ( 2008 ) . also see ghosh and rao ( 1994 ) , rao ( 2003 ) , datta and ghosh ( 2012 ) and pfeffermann ( 2013 ) for a good survey on this topic .  in the most of literatures related to the fay - herriot model or the nested error regression model , it is assumed that the data has real values .however , we are often faced with the data with positive values like the price data and fitting such data to the normal distributions in nested error regression model may be inappropriate in the case that distribution of data is skewed .moreover , if we analyze bounded data like ratio data as real - valued data , we may cause the problem that prediction intervals of model - based predictors may be out from the range of the bounded data .thus , this paper is focused on developing a methodology for analyzing positive or bounded data by incorporating a parametric transformation into the conventional nested error regression model .a standard approach to analyzing positive data is to apply the log - transformation to the data , which was investigated in slud and maiti ( 2006 ) in the nested error regression model .however , such a specific transformation is not always appropriate , and we want to adjust the transformation flexibly to fit the transformed data to a normal linear mixed regression. a conventional method in this direction is the box - cox transformation suggested by box and cox ( 1964 ) , described by however , it is known that the maximum likelihood estimator of the transformation parameter in the box - cox transformation is not consistent , so that the resulting eblup is not consistent to the best linear unbiased predictor ( blup ) .thus , the inconsistency in the estimation of is a crucial problem in the context of small area prediction . as an alternative transformation ,in this paper , we use the dual power transformation ( dpt ) ( yang , 2006 ) which will be described in section [ tnerm ] .this is a transformation from positive numbers to real numbers , and it can be expected that the maximum likelihood estimator of is consistent . for analyzing bounded data , we propose the dual power logistic transformation ( dplt ) which transforms the bounded data to real - valued data .this transformation will be also described in section [ tnerm ] .  utilizing these transformations , in this paper , we suggest the parametrically transformed nested error regression model ( tnerm ) defined as where s and s are mutually independently normally distributed and is a fixed effect . here , is a general transformation function which is characterized with the transformation parameter .the examples of treated in this paper are dpt and dolt .the transformation parameter can be used for adjustment , and the proposed model enables us to flexibly analyze the small - area positive or bounded data . the detailed model description is given in section [ tnerm ] .  in the conventional nested error regression, we predict the quantity , where by the empirical best linear unbiased predictor .sugasawa and kubokawa ( 2014 ) proposed the parametrically transformed fay - herriot model and the gave second - order unbiased estimators of mse of .however , in the transformed model , the quantity of interest is the inversely transformed function rather than .thus , in this paper , we consider to predict and propose the transformed empirical best linear unbiased estimator ( teblup ) , namely . since this predictor is expected to give reliable predicted values for small - areas with higher precisions , it is important to assess uncertainty of teblup . in the context of small area estimation , we have two approaches to measuring the uncertainty : one is to provide an estimate of the mean squared error ( mse ) of the prediction ( see datta and lahiri , 2000 , datta , 2005 and hall and maiti , 2006a , b ) , and the other is the prediction ( confidence ) intervals ( see basu , , 2003 , chatterjee , 2008 , datta , 2002 , diao , 2014 and yoshimori and lahiri , 2014 ) .  the goal of this paper is to construct prediction intervals of based on .since it is harder to derive an analytical prediction interval with suitable accuracy based on the taylor series expansion , we here provide a prediction interval with second - order accuracy based on the parametric bootstrap along the line given in chatterjee , lahiri and li ( 2008 ) .we also provide a conditional prediction interval given data in the area of interest , motivated from the results of booth and hobert ( 1998 ) who discussed a conditional mse and its estimation in generalized linear mixed models .the paper is organized as follows : in section [ tnerm ] , we suggest the parametric transformed nested error regression model with dpt or dplt. some consistent estimators of parameters in tnerm are also given .in section [ prediction ] , we introduce the transformed empirical best linear unbiased predictor and construct unconditional and conditional prediction intervals with second order accuracy based on the parametric bootstrap method . in section [ num ] , we conduct simulation studies . in section [ emp ] , we apply the proposed model to two data set , the survey data in japan and crop areas data given in battese ( 1988 ) .the concluding remarks are given in section [ remark ] .all the technical proofs are given in the appendix .consider the two - stage cluster sampling , namely , clusters are randomly selected , and data are randomly selected from each selected cluster . for , a random sample taken from the -th cluster with size denoted by .the most useful model for analyzing such data is the nested error regression model ( nerm ) described by where s and s are mutually independently distributed as and . here, a vector denotes the transpose of , denotes a normal distribution with mean and variance , is a -dimensional known covariate associated with , is a -dimensional unknown vector of regression coefficients , and and are unknown components of variance , called ` between and ` within components , respectively .the model ( [ nerm ] ) is a linear mixed model which incorporates both fixed and random effects , and it has been used for analyzing unit level data in the framework of small - area estimation . when s are real - valued data , the model ( [ nerm ] ) is reasonable .however , it is not necessarily appropriate when values of s are limited to spaces of positive or bounded numbers. then , we need to consider a transformation of to fit into nerm . in this paper , we consider two types of transformations for the data limited to or , where and for the real space . for , namely positive data , we use the dual power transformation ( dpt ) suggested by yang ( 2006 ) , described as for .it is noted that for , the inverse transformation is expressed as for , and for . when data are restricted on the space for , dpt can be extended to for analyzing data on the space . for , we newly suggest the dual power logistic transformation ( dplt ) given by for . using the expression of the inverse transformation of dpt ,one gets the inverse transformation of dplt , given by for , and for .when data are restricted on the interval for fixed values and , , dplt can be extended to , since lies in .thus , we can analyze data on using .let be or for given .the transformation parameter is adjusted so that transformed data s can fit into nerm .thus , we can suggest the parametrically transformed nested error regression model ( tnerm ) it may be convenient to write the model ( [ model ] ) in matricial forms .let , , and . also , define by then , the model ( [ model ] ) is expressed as and has an -variate normal distribution where for , the identity matrix and .it is noted that the covariance of has the intra - class correlation structure , namely are not mutually independent when .let .all the data s are described as the -dimensional vector .then the joint density function is expressed as where is the jacobian of the transformation for .this expression will be used for estimating the unknown parameters , , and .we here provide consistent estimators of the unknown parameters , , and . to this end , we begin by estimating , and in the case of known . in this case , the conventional estimators given in the literature for nerm ( [ nerm ] ) can be inherited to the transformed model .  concerning estimation of , the maximum likelihood ( ml ) or generalized least square ( gls ) estimator of for known and is where , is the mean of covariates s for the -th area , and is the mean of the transformed observations . since , it is clear that is consistent and under the following assumption :  [ barx ] the following are assumed for and :  * converges to a positive definite matrix as .* there exist integers and which are positive and independent of such that for .  since and are unknown , we estimate them and substitute their estimators into . in nerm ( [ nerm ] ) with known , for estimation of and , the prasad - rao estimator , the maximum likelihood ( ml ) and the restricted maximum likelihood ( reml )estimators have been used in the literature , and it would be plausible that those estimators can be used still in tnerm ( [ model ] ) by replacing with an estimator .we here clarify conditions that estimators of and should satisfy in order to derive prediction intervals given in this paper . for notational convenience , means that every component in is of order , and the notation is defined similarly .[ var ]  let be an estimator of in the case of known . then it is assumed that the estimator satisfies the following :  * .* =\o_p(m^{-1}) ] .condition ( a.3 ) implies that the estimators and are consistent .conditions ( a.4 ) and ( a.8 ) will be used for investigating asymptotic properties of .  substituting into ,one gets the estimator defined by it is noted from ( a.8 ) that .some asymptotic properties on are given in the following lemma which will be proved in the appendix .lemma [ hbeta ] will be used in theorem [ thm.boot ] for showing the second - order accuracy of the parametric bootstrap procedure .[ hbeta ] under assumptions [ barx ] and [ var ] , it holds that , =\o_p(m^{-1}) ] .based on the results given in the above lemmas , we can get the following asymptotic properties of estimators for the unknown parameters in tnerm ( [ model ] ) . the proof is given in the appendix .[ paraest ] let and . under assumptions [ barx ] and [ var ] , we have and for .  the latter property that is technical but crucial for the proof of theorem [ thm.boot ] in section 3 , which gives validity of the bootstrap method for constructing prediction intervals of teblupwe now provide the transformed empirical best linear unbiased predictor ( teblup ) for small area estimation and construct the prediction intervals based on teblup as a measure of uncertainty of the predictor . since teblup includes the estimators of the parameters , , and , it is difficult to construct an exact prediction interval .thus , in this section , we try to construct a prediction interval with the second - order accuracy . to this end, the asymptotic results given in the lemmas in the previous section are heavily used .we here consider the problem of predicting the quantity when is known , it is well known that the conditional distribution of given is , where = \barx_i'\bbe+\frac{n_i\rho}{1+n_i\rho}(z_i(\la)-\barx_i'\bbe),\ ] ] and the estimator is the bayes estimator of in the bayesian context . substituting the gls given in ( [ beta ] ) into ( [ mut ] )yields the predictor it is known that this estimator is the best linear unbiased predictor ( blup ) of . for , and , we substitute the estimators given in section [ tnerm ] into the blup , and the resulting predictor is given by where , for simplicity , we use the notations , , and as abbreviation of , ,  without any confusion .the predictor ( [ eblup ] ) is called the empirical best linear unbiased predictor ( eblup ) . in the bayesian context, it corresponds to the empirical bayes estimator of .since our interest is in the prediction of , we need to make the inverse transformation of .it should be remarked that the inverse transformation depends on the unknown transformation parameter .hence , the transformed predictor of is given by which is called the transformed empirical best linear unbiased predictor ( teblup ) .for measuring uncertainty of teblup , we propose prediction intervals of with a second - order accuracy for .the basic idea of constructing prediction intervals are based on chatterjee , ( 2008 ) , who proposed the parametric bootstrap method for constructing a second - order accurate unconditional confidence interval in normal linear mixed models .recall that conditionally , where and are given in ( [ mut ] ) and ( [ sigt ] ) , respectively .the conditional distribution given implies that is a standard normal pivot since .let . for given in ( [ eblup ] ) , we want to obtain a distribution of this distribution is denoted by . if there were constants and such that =1-\al ] .this conditional mse measures how much the eblup has an estimation error given the data , and this conditional approach may be appealing because it conditions on the data in the area of interest . in this subsection, we construct a conditional prediction interval given such that to this end , we need to approximate the conditional distribution of given .denote this conditional distribution by .the difference between the unconditional and conditional prediction intervals is that the unconditional distribution of is considered in ( [ ed ] ) , while the conditional distribution of given is treated .it is noted that there is a correlation between and in ( [ cp ] ) , namely , the conditional distribution of given is for and given in ( [ mut ] ) and ( [ sigt ] ) .since it is difficult to derive an exact conditional distribution of given , we suggest to approximate it via the parametric bootstrap method . a bootstrap sample is generated as where s and s are mutually independently distributed as and .let for .noting that is fixed , we can construct the estimator from with the same technique as used to obtain .let and for and .let be a random variable having for .then , for fixed , we consider the distribution of which is denoted by .similarly to the unconditional case , we can obtain a conditional prediction interval via the parametric bootstrap approximation .[ thm.boot.cond ] under assumptions [ barx ] and [ var ] , we have   the proof of theorem [ thm.boot.cond ] is given in the appendix . from theorem [ thm.boot.cond ], we obtain a conditional prediction interval with second - order accuracy .[ cor.boot.cond ] for any , let and be appropriate quantiles based on the bootstrap sample such that where is the distribution function of .then , one gets the prediction interval of given by .\ ] ] under assumptions [ barx ] and [ var ] , it holds that this section , we investigate performances of the procedures suggested in the previous sections by monte carlo simulation .we investigate disadvantages when we use a misspecified transformation under dpt or dplt .+ * [ tnerm with dpt ] *  since the purpose of small area estimation is to predict the mean of each small area as accurately as possible , we begin by comparing the prediction error among tnerm with dpt , the log - transformed nerm and the non - transformed nerm .we generate observations from the model for and , where and are generated from with and with , respectively , are generated from , which are fixed through simulation runs , and .we consider the 6 patterns of .we note that the log - transformation is a misspecified transformation when , and the identity transformation is always misspecified regardless of .the true values we want to predict is let be the teblup defined in ( [ teblup ] ) and let and be predictors based on the log - transformed nerm and the non - transformed nerm , respectively. then we can define the prediction mean squared error ( pmse ) as , \ \ \ \ k=1,2,3\ ] ] which can be estimated based on simulation runs . to see the differences among pmse , we calculate the improvement ratio of pmse ( irp ) defined as simulation results are given in table 1 .it is observed that pmse in log - nerm gets worse than that in tnerm when is away from . when , the log - transformation is the true transformation , so that it is natural that pmse in tnerm is slightly larger than that in log - nerm since there is an estimation error of in tnerm . in the conventional non - transformed nerm ,the prediction errors are always bad compared to the other two models , but gets better as gets larger .this is because the dpt is similar to the identity transformation when is close to .we next investigated the percentage of zero estimates of , variance of random effects .zero estimates of means that the resulting eblup estimator of a small area mean is over - shrunk to the regression estimator .such a situation is not preferable in practice and recently li and lahiri ( 2010 ) and yoshimori and lahiri ( 2014 ) discussed this problem .based on the same simulation runs , we calculate the percentage of zero estimates of .we report the simulation result in table 1 .the percentage in log - nerm is slightly larger compared to tnerm , and the percentage in conventional nerm are always large but gets smaller as goes to , which seems to be the same reason in the prediction error ..the percentage ratio of prediction errors in tnerm with dpt compared to log - transformed nerm and nerm , and the percentage of zero estimates of for three models . [cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ]in this paper , we have suggested the parametric transformed nested error regression model ( tnerm ) as a new unit - level model for analysis of positive or bounded small area data .we have provided the procedures for estimating unknown parameters including the transformation parameter as well as regression coefficients and the variance components .as parametric transformations , we consider the dual power transformation for positive data and the dual power logistic transformation , which we newly proposed in this paper , for bounded data .the transformed eblup ( teblup ) has been made based on the consistent estimators , and unconditional and conditional prediction intervals with second - order accuracy have been constructed based on the parametric bootstrap method . through the simulation , it is confirmed that prediction accuracy can be bad and the percentage of zero estimation of tends to be large under misspecified transformation , which motivated us to take the transformation parameter into account of model formulation .the finite sample performances of proposed prediction intervals have been confirmed by simulation as well and the coverage probability of the suggested prediction intervals is close to the nominal level . for real data applications , we applied tnerm to survey ( positive ) data in japan and famous crop areas ( bounded ) data treated in battese , ( 1988 ) .the crucial properties of dpt and dplt for giving validity of proposed methodology in this paper are summarized in the appendix and we can use another parametric transformation as alternative to dpt or dplt whenever it holds the properties .our proposed methodology based on the parametric transformation is regarded as a new framework to cope with small - area data , and we hope further development will be studied from theoretical and practical aspects in statistical inferences .     for notational convenience ,let for be the partial derivative of , i.e. . for example , , and others .moreover or for means that or respectively .  in their proofs, the following fact will be heavily used : assume that for , a function is independent of for , and that and =o(1) ] , then from the central limit theorem ( clt ) , one gets \bigr ) \big|\y_i = o_p(1 ) , \ \ \ i=1,\ldots , m,\ ] ] where denotes a random variable given . in the proofs , for notational simplicity , we treat the case of without loss of generality .     *  is a monotone increasing function of ( ) and its range is , where is the domain of transformation , namely for dpt and for dplt .*  and are three times continuously differentiable , where is the inverse function of defined by . * the moments of the following exist for each fixed : + ( 1 ) and , + ( 2 ) , and , + ( 3 ) , , , , and , + where their expectations are taken with respect to which is normally distributed .property ( p.1 ) means that the transformation is a one - to - one and onto function from to .clearly , ( p.1 ) is not satisfied by the box - cox transformation ( box and cox , 1964 ) , but by the logarithmic transformation .properties ( p.2 ) and ( p.3 ) will be used for establishing consistency of estimators including transformation parameter and for constructing prediction intervals .especially , ( p.2 ) and ( p.3 ) ( 1 ) guarantees that the random variable given in ( [ eq.la ] ) converges in probability , and ( p.3)(2 ) guarantees that converges in probability .recall that .it is noted that here is expressed as where .note that since from ( [ clt ] ) and assumption [ barx ] .thus , .also , can be expanded as which implies that and =o_p(m^{-1}) ] , from ( [ a1.1 ] ) , it is sufficient to show that =o_p(m^{-1}) ] , it is seen that from ( [ clt ] ) .thus , , namely , also , we obtain which implies that since and it depends only on of , from ( [ lln ] ) and ( [ clt ] ) , one gets \big)\bigr|\y_m = o_p(m^{-1/2}).\ ] ]  we next evaluate .we here give a proof for , and we omit proofs for the other elements since they can be similarly proved . by a straightforward calculation , where which is since as in the proof of lemma [ hbeta ] .then , so that , we have since =\sige+n_i\sigv ] and  ] . since is continuous , without loss of generality, we have and for some positive . then , and converge to and , respectively , in probability .this implies that both probabilities and converge to one as .in fact , for instance , the former result follows from the fact that as since .thus , for any , there exists an such that for any , and .note that the intersection of the events and implies that is included in the interval , namely , .hence , which means that is consistent .we next show that in the case of . to this end , we expand the equation ( [ eq.la ] ) around to get where is an intermediate value between and . for the numerator in ( [ consist ] ) , from lemma [ hbeta ] and assumption [ var ] , it is seen that where for and the density function of .since are mutually independently distributed with =0 ] , where from ( [ lln ] ) and assumption 1 , we have so that .since is an matrix , it follows that .these observations show that the denominator in ( [ consist ] ) is of order .hence , one gets , namely , .finally , we show that =o_p(m^{-1}) ] , which is of order . from lemma [ hbeta ] and assumption [ var ], it easily follows that then , one gets , so that =-{m}^{-1}e[f(\hbbe(\la),\hsigv(\la),\hsige(\la),\la)]+o(m^{-1}).\ ] ] note that is evaluated as from assumption ( a.4 ) , it is easy to see that =o_p(m^{-1}) ] , which conclude that =o_p(m^{-1}) ] .then , one gets and = \o_p(m^{-1}) ] . therefore , the proof is complete .it is first noted that in the proof , the capital , with or without suffix , means a generic constant .if is expanded as where is a smooth function with , then the corresponding expansion holds for , namely , thus , one gets which establishes the result given in theorem [ thm.boot ] .hence , we shall show the expansion ( [ a5.1 ] ) through the following steps .since the inequality for any is equivalently rewritten as , we have \\ & = e\bigl(p\bigl[\si_i^{-1}(\xi_i-\xih_i(\bth))\leq \si_i^{-1}\bigl\{h(h^{-1}(\xih_i^{eb}+q\sih_i,\hla),\la)-\xih_i(\bth)\bigr\}\bigr]\bigr| \y\bigr)\\ & = e\bigl [ \phi(q+r(q,\y))\bigr],\end{aligned}\ ] ] where is a cumulative distribution function of the standard normal distribution and for the standard normal density function , the first and second derivatives are written as for .the taylor expansion is applied to get -\frac12q\phi(q)e[r^2(q,\y)]\\ & + \frac12e\bigl[\int_q^{q+r(q,\y)}\bigl ( q+r(q,\y)-x \bigr)^2(x^2 - 1)\phi(x)dx\bigr ] \\ & = \phi(q)+\phi(q)t_1(q)-\frac12q\phi(q)t_2(q)+t_3(q),\end{aligned}\ ] ] where  ] and  ] from ( [ r ] ) . to this end , decompose as , where from ( [ q1 ] ) , is expanded as where and .it is here noted that =o_p(m^{-1 } ) , \label{a5.5}\ ] ] which will be shown in ( step 4 ) below .then , it follows that the last two terms of the expansion of are given , and by the similar argument .thus , =h_x^{-1}(\xih_i(\bth)+q\si_i,\la)e(u|\y_i)+o_p(m^{-1 } ) = o_p(m^{-1}).\ ] ] also , is expanded as where is intermediate value between and . it can be observed that , under assumption [ trans ] and from lemma 4 .thus , and &=e\bigl [ h^{-1}_{\la}(\xih_i^{eb}+q\sih_i,\la)(\hla-\la ) \bigr|\y_i\bigr]+o_p(m^{-1})\\ & = e\bigl[\bigl\{h^{-1}_{\la}(\xih_i^{eb}+q\sih_i,\la)-h^{-1}_{\la}(\xih_i(\bth)+q\si_i,\la)\bigr\}(\hla-\la)\bigr|\y_i\bigr]\\ & \ \ \ \ + h^{-1}_{\la}(\xih_i(\bth)+q\si_i,\la)e(\hla-\la|\y_i)+o_p(m^{-1})\\ & = o_p(m^{-1}),\end{aligned}\ ] ] since given is , which can be verified by and lemma [ paraest ] .it remains to show that and , for which it is sufficient to show that both and are and the conditional expectation given is .recall that .first , is rewritten as note that given , and further , from lemma [ paraest ] and a similar expansion as in ( [ hr ] ) , it follows that and .hence , one gest and \\ & = e\bigl[\frac{\rho n_i}{1+\rho n_i}z_{i,\la}(\la)(\hla-\la)+\frac{n_i}{(1+\rho n_i)^2}(\hr-\rho)(z_i(\la)-\barx_i'\bbe)\bigr| \y_i\bigr]+o_p(m^{-1})\\ & = \frac{\rho n_i}{1+\rho n_i}z_{i,\la}(\la)e[\hla-\la|\y_i]+\frac{n_i}{(1+\rho n_i)^2}(z_i(\la)-\barx_i'\bbe)e[\hr-\rho| \y_i]+o_p(m^{-1}),\end{aligned}\ ] ] which is of order from lemma [ paraest ] .a similar evaluation for shows that given , then , from lemma [ paraest ] , it follows that and =o_p(m^{-1}) ] , we have an asymptotic expansion of as for an smooth quantity , which leads to the result by lemma 4 .chatterjee , s. , lahiri , p. and li , h. ( 2008 ) .parametric bootstrap approximation to the distribution of eblup and related predictions intervals in linear mixed models ._ , * 36 * , 1221 - 1245 .datta , g. s. , ghosh , m. , smith , d. d. and lahiri , p. ( 2002 ) . on an asymptotic theory of conditional and unconditional coverage probabilities of empirical bayes confidence intervals ._ , * 29 * , 139 - 152 .diao , l. , smith , d. d. , datta , g. s. , maiti , t. and opsomer , j. d. ( 2014 ) .accurate confidence interval estimation of small area parameters under the fay herriot model . __ , * 41 * , 497 - 515 .

Abstract: for analyzing positive or bounded data , this paper suggests parametrically transformed nested error regression models ( tnerm ) , which transform the data flexibly to follow the normal linear mixed regression . as useful transformations , we consider the dual power transformation for positive data and the dual power logistic transformation newly proposed for bounded data . we provide a procedure for estimating consistently the parameters of the proposed model and a predictor based on the consistent estimators . then , in order to calibrate uncertainty of the transformed empirical best linear unbiased predictor , we derive both unconditional and conditional prediction intervals with second - order accuracy based on the parametric bootstrap method . the proposed methods are investigated through simulation and empirical studies .   dual power transformation , linear mixed model , nested error regression model , parametric bootstrap , prediction intervals , small area estimation .


assume that a source wants to leak top secret documents anonymously .it is considered secure to do so through tor using a privacy - enhancing browser plugin such as torbutton .however , assume that , at the same time , this source uses another insecure application on tor .is it then possible to associate the top secret documents with the ip address of the anonymous source ?the answer to this question is yes !  by exploiting tor s design , one can indeed exploit an insecure application to associate the usage of a secure application ( e.g. , the one leaking top secret documents ) with the ip address of a tor user .this attack against tor consists of two parts : ( a ) exploiting an insecure application to reveal the source ip address of , or _trace _ , a tor user and ( b ) exploiting tor to associate the usage of a secure application with the ip address of a user ( revealed by the insecure application ) .as it is not a goal of tor to protect against application - level attacks , tor can not be held responsible for the first part of this attack .however , because tor s design makes it possible to associate streams originating from secure application with traced users , the second part of this attack is indeed an attack against tor .we call the second part of this attack the _ bad apple attack_. ( the name of this attack refers to the saying `` one bad apple spoils the bunch . ''we use this wording to illustrate that one insecure application on tor may allow to trace other applications . )this paper differs from the related work in three main aspects .first , we launched our attacks on the real tor network for a substantial period of time and revealed 10,000 ip addresses of `` anonymous '' tor users .to the best of our knowledge , it is the largest attack against the tor network in number of revealed ip addresses .second , whereas most attacks against tor were targeted to web browsers , we directly target p2p filesharing applications ( i.e. , bittorrrent ) in this study .bittorrent traffic generates a significant fraction of tor traffic in volume ( more than 40% ) , making it a primary target for attackers .in addition , we show that 70% of bittorrent users on tor establish p2p connections _ outside _ of tor , making most bittorrent tcp connections ( and traffic ) invisible to the tor network .thus , the number of bittorrent users on tor is likely to be largely underestimated and so more tor users are susceptible to our attacks .third , whereas the principle of the bad apple attack has been discussed in the past , it is an open question whether it allows to trace a significant number of streams originating from secure applications . actually , that many tcp streams can share one circuit is listed as the fourth _ improvement _ of tor over the old onion routing design because it is supposed to `` improve efficiency and anonymity . ''we note that roger dingledine cited an initial version of this work to confirm the need to `` brainstorm about ways to protect users even when their applications are handing over their sensitive information '' on the website of the tor project .the main contributions of this paper are as follow :  * we design two attacks against bittorrent to reveal the ip address of bittorrent users on tor .* we instrument six tor exit nodes and launch our attacks on the real tor network for a period of 23 days .we reveal 10,000 ip addresses of `` anonymous '' tor users . *we show that the bad apple attack allows us to trace 193% of additional streams as compared to bittorrent streams , including 27% of http streams . in total, we traced 9% of _ all _ tor streams carried by our instrumented exit nodes .* we profile bittorrent and web usage on tor per country of origin ( which would be impossible without first tracing tor users ) . we show that users on tor are over - represented in some countries as compared to users that do not use tor . by analyzing the type of content downloaded , we explain this behavior by the higher concentration of pornographic content downloaded at the scale of a country . we present results that suggest the existence of an underground ecosystem on tor .the rest of this paper is organized as follow .first , we briefly discuss the ethical and legal considerations of running attacks against production systems in section [ sec : ethics ] .we then give an overview of tor and explain how this design can lead to the bad apple attack when an insecure application reveals the source ip address of a tor user in section [ sec : onion ] . in section [ sec : attacks ] , we discuss our attack model , two attacks against , and how the bad apple attack specifically applies to .we show that the bad apple attack is severe enough to profile not only downloads but also the websites visited by tor users in section [ sec : profil ] .we discuss the related work in section [ sec : related ] .finally , we summarize our contributions and give a few perspectives in section [ sec : discuss ] .in order to comply with the legal and ethical aspects of privacy , we performed our analysis on the fly and do not store any nominative information such as ip addresses .we logged only the asn and country of traced tor users to be able to perform this study .in addition , we only present aggregated statistics as suggested by loesing et al . in  .finally , we have also been cautious not to inadvertently dos tor or infrastructures , or interfere with the normal usage of those systems .tor is a low - latency anonymity network .as stated in the original paper , its main design goals are to prevent attackers from linking communication partners and from linking multiple communications to or from a given user .tor relies on an overlay network and on onion routing to anonymize tcp - based applications like web browsing and p2p filesharing .tor explicitly made the design choice to support _ only _ tcp which `` helped portability and deployability '' .when a client communicates with a server via tor , she selects nodes of the tor system ( where is typically 3 ) and builds a circuit using those nodes .messages are then encrypted times , first with the key shared with the last node ( called _ exit node _ ) of the circuit , and subsequently with the shared keys of the intermediate nodes from to . as a result, each intermediate node only knows its predecessor and successor , but no other nodes of the circuit .in addition , only the exit node is able to recover the original message .  to improve efficiency , tor multiplexes several streams from the same source into a single circuit .originally , onion routing used a separate circuit for each stream but it required multiple public key operations for every request .it has also been argued that creating many circuits degraded privacy because it implied to contact more tor nodes , some of which may be compromised . however, we will show that , when several streams are multiplexed into a single circuit , a single stream whose source ip address is revealed allows an attacker to associate many additional streams with the same traced user .[ [ section ] ]  is a popular peer - to - peer ( p2p ) protocol for file replication . to download a content, a client first discovers peers sharing that content using centralized trackers , a distributed tracker ( dht ) , and peer exchange ( pex ) ._ trackers _ are servers storing content identifiers and for each identifier , a list of peers distributing the corresponding content .a peer subscribes its ip / port to the tracker for a given content identifier and requests a list of peers for that content when it starts downloading and then periodically after that ( e.g. , every 10 minutes ) .communications with trackers is typically done in clear over tcp ( i.e. , through tor ) therefore _ a malicious exit node can tamper with the lists of peers returned by centralized trackers . _ ( furthermore , centralized tracking is sometimes also done over udp with consequences similar to those we discuss after . )in addition to centralized trackers , clients can also use a decentralized tracker based on a distributed hash table ( _ dht tracking _ ) .whereas dht tracking is often used in combination to centralized tracking , it can also be used alone with magnet links .mainline dht tracking works as follow .first , a client ( dht tracker ) picks an identifier that is coded on the same space as content identifiers .then , a peer interested in downloading a content uses the content identifier to locate the corresponding dht tracker after which , it subscribes its ip / port to that tracker and requests a list of peers ( just as with a centralized tracker ) .communication with dht trackers is done over udp therefore _ a tor user may subscribe her public ip / port to the dht tracker . _  as a client discovers peers , it tries establishing tcp connections and if successful , sends an application handshake containing the content identifier and in the case of an extended handshake , the listening port number . that p2p connection is also used for content distribution .whether a p2p connection is established through tor has a tremendous impact on performance and can be configured by the user therefore _ a tor user may establish p2p connections using her public ip / port ._  finally , after using centralized or dht tracking , more peers can be discover using peer exchange ( _ pex _ ) .with pex , users typically exchange lists of peers they are connected to over established p2p connections therefore _ a tor user may subscribe her public ip / port to her pex partners . _all our attacks require to control one exit node in order to trace its tor users . from january to february 2010 ( 23 days ) , we instrument and monitor six tor exit nodes spread throughout the world ( two in asia , two in europe , and two in the u.s ) and launch the attacks described after . the first attack , the hijacking tracker s responses , also requires to control a peer publicly connectable so it can accept incoming tcp connections and receive handshake messages . for practical reasons, we performed this hijacking attack on only one of our exit nodes .hijacking tracker s responses exploits the fact that centralized tracking is done through tor and the list of peers can be tampered with by a malicious exit node .one condition for this attack to trace users is that they do content distribution outside of tor .we will see in section [ sec : hijack ] that this is the case for 70% of the users on tor .the second attack exploits the fact that dht tracking uses udp and so is done outside of tor .we note that if dht tracking was instead done over tcp through tor , it would still be possible to perform an hijacking attack as with centralized tracking .we suspect that it is possible to perform similar statistical attacks with pex and centralized udp tracking , however , we did not exploit them in this study .     of peersuse tor only to connect to the tracker . _hijacking the tracker responses consists in inserting the ip / port of a peer controlled by the attacker ( malicious peer ) into the list of peers returned by the tracker so the targeted user connects to the malicious peer .when peers use tor only to connect to the centralized tracker and not to distribute content , they will connect to the malicious peer directly , i.e. , outside of tor , allowing the attacker to trace them . but how can the attacker distinguish between a direct connection from one that is going through tor ?this can be done by collecting the ip addresses of tor exit nodes ( which are public ) to check whether an incoming connection at the malicious peer originates from one of these addresses .if it does , the targeted user is distributing content through tor . otherwise , he uses tor only to connect to the tracker and the connection to the malicious peer is direct ( the source ip address field of the ip datagrams contains the real ip address of the targeted user ) .  in fig .[ fig : bt_usage ] , we observe that most users use tor only to connect to the centralized tracker , making the hijacking of tracker s responses a simple yet efficient attack .one explanation for this behavior may be that users distribute content outside of tor to not degrade performance .in addition , piatek et al . have showed that naive spies use tracker subscriptions as evidences of copyright infringement .this might be another reason why users are mainly concerned by anonymizing their tracker subscription .  because most p2p connections are established _outside _ of tor , most bittorrent streams ( and traffic ) are invisible to the tor network .thus , the number of bittorrent users on tor is likely to be largely underestimated and so more tor users are susceptible to our attacks .the second attack exploits dht tracking .because dht tracking is carried over udp ( which tor does not support ) a client fails to connect to the dht through tor and reverts to using its _ public _ interface to publish his ( public ) ip address and listening port into the dht .say the targeted user is downloading content1 .the ip address and listening port of that user are then stored on the peer responsible of tracking content1 in the dht . but how can an attacker in tor locate the peer storing the public ip address of the targeted user ? andthen , how to distinguish the ip address of the targeted user from the other ip addresses downloading content1 ?  to find the information of a targeted user in the dht , we use the content identifier and listening port number contained in the subscription to the centralized tracker and extended handshake messages . when one of our exit nodes receives one of these messages , it immediately locates the peer tracking that content identifier and collects all the ip / port couples that have subscribed for that identifier .all peers who have subscribed for that content identifier are candidates to be associated with the tor user .we validate that the listening port is a good identifier of a peer within a torrent in fig .[ fig : dht_valid ] .we then associate the ip address of the only peer with a matching listening port to the targeted user . if there is no such peer or that there is more than one , we consider that we have failed to trace the targeted user .now that we have seen how to trace streams , we describe the specifics of the bad apple attack for users .as we have already discussed , the bad apple attack can reveal the source ip address of streams in the same circuit , or even different circuits . with as with any other application , revealing the source ip address of streams in the same circuitis straightforward ; when the source ip address of a stream is revealed , all streams multiplexed into the same circuit are associated with the same traced user .  to reveal the source ip address of streams in different circuits , we exploit two patterns in signalling traffic .the first pattern is the peer identifier which is essentially a random string of 20 bytes .thus , when we observe the same peer identifier in the messages of different circuits , we consider that these circuits have the same source .one problem with this first pattern is that it does not work when communication between peers is encrypted . to alleviate this issue, we also consider communication to an ip / port freshly returned in a tracker response as a second pattern .in particular , if in a circuit we observe that a peer initiates communication with an ip / port contained in a tracker response from another circuit , we link the two circuits . we note that the linkage of streams in different circuits is particularly severe because , when an attacker traces a tor user , he can potentially associate past or future circuits with that user , without need to reveal his ip address a second time .we found that the bad apple attack applied to allowed us to trace 193% of additional streams as compared to bittorrent streams , including 27% of http streams possibly originating from `` secure '' browsers .we show the number of non - bittorrent streams traced per destination port number in fig .[ fig : bad_onion ] .  [ cols="^,^,^,^,^",options="header " , ]by analyzing the traffic relayed by our exit nodes , we evaluate that 19% of all streams on tor are bittorrent streams .( we remark that this percentage is much higher than in mccoy et al . , suggesting that the number of bittorrent users on tor has increased since 2008 . )we successfully trace 9% of _ all _ tor streams . in this section ,we use the resulting 10,000 traced ip addresses to profile the bittorrent downloads and websites visited by tor users per country of origin .we start by investigating whether bittorrent utilization per country and as on tor is different relatively to bittorrent utilization outside of tor .we then analyze the content downloaded by bittorrent users on tor for a few interesting countries and investigate the existence of an underground bittorrent ecosystem on tor .finally , we analyze the websites visited by tor users per country of origin .to compare bittorrent utilization on tor and outside of tor , we need a representative sample of bittorrent users for each of these two utilizations . we see in fig .[ fig : stability ] that , after 10 days , most of the top10 countries and as ( in number of ip addresses of traced tor users ) have reached their final rank . therefore , we have a reasonably representative sample of the utilization of bittorrent _ on tor _ in these countries and ases .  as a representative sample of the utilization of bittorrent _ outside of tor _ , we then use a sample of 10,000,000 ip addresses collected on august 22nd 2009 on the piratebay , the largest bittorrent tracker at that time .indeed , the piratebay was an order of magnitude larger than the second largest bittorrent tracker at the time of the measurement therefore we argue that a daily sample from the piratebay is reasonably representative of the global utilization of bittorrent outside of tor .we refer to le blond et al . for a description of the measurement methodology to collect this data .table [ tab : rank - top10-country ] shows the popularity of users on tor per country ( left ) and as ( right ) . the over - representation ( over ) for a given country ( resp . as )is the fraction of ip addresses on tor in that country ( resp . as )divided by the fraction of ip addresses outside of tor in the same country ( resp . as )we do not show the over - representation in china because chinese content are not generally tracked by the piratebay , the tracker that we have used to capture the location of bittorrent users outside of tor .hence , we greatly over - estimate the over - representation in china .an over - representation of in the us means that there is about the same fraction of us users on tor as outside of tor . andan over - representation of in japan means that there are times more users from japan on tor than outside of tor . in other words , whereas us users do not hide on tor more than average , japanese users strongly do .the reasons behind such behavior may be technological , political , sociological , etc .[ [ sec : bt_downloads ] ] explaining over - representations + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +      to investigate the over - representations observed in section [ sec : country ] , we now analyze the types of content downloaded by tor users from countries with very different over - representations .in particular , we had observed that us users were not over - represented on tor whereas japanese and german users were . in fig .[ fig : bt_profile ] , we indeed see that us users are downloading a large variety of content as compared to japanese users who mainly download hentai ( pornographic animes ) , and german users who mainly download pornographic movies . therefore , we argue that the reasons for over - representations ( at least in bittorrent ) are mainly sociological .bittorrent comprises communities of users , or _ ecosystems _ ,among which content are distributed , , .there is one _ public ecosystem _ that can be accessed by all internet users , and several _ private ecosystems _ with many users whose access is restricted to registered users .even though private ecosystems are more difficult to monitor than the public ecosystem , registration is generally open to everyone and a single registered user can in principle monitor the whole community .because even private ecosystems are indeed relatively easy to monitor , it is probable that some private ecosystems be known only by peculiar members ( e.g. , downloaders of child pornography ) thus forming an _ underground ecosystem_.  to investigate the existence of such an underground ecosystem on tor , we check whether there is a subset of the content distributed on tor that belongs neither to the public nor private ecosystems .we use .torrent files from 7,110,000 bittorrent content : 6,800,000 from the public bittorrent ecosystem , and 310,000 from the private bittorrent ecosystem .we also search the missing .torrent files on google . to the best of our knowledge ,this is the largest collection of .torrent files ever assembled .we see in fig .[ fig : bt_public ] that 3% of all the content distributed by bittorrent users on tor belong neither to the public nor private ecosystems .this result suggests the existence of an underground ecosystem on tor .however , one would need to download these unknown content and to check them manually in order to determine whether they belong to regular private ecosystems for which we do not have the .torrent files or whether they are more sensitive .we have showed that the bad apple attack allowed us to trace 193% of additional streams as compared to bittorrent streams , including 27% of http streams possibly originating from `` secure '' (i.e. , proxied ) browsers . in the following ,we use the web - filtering service of fortiguard to analyze the type of websites visited on tor per country of origin .we show the type of websites visited per country of origin in fig .[ fig : http_usage ] . because they all are bittorrent users , traced users differ from the average tor users ( _ tor _ ) .in particular , around 50% of the requests are targeted to filesharing websites ( _ filesharing _ ) such as _ thepiratebay _ , _ megadownload _ , or _rapidshare_. traced tor users also visit significantly more hacking websites suggesting that they are interested in security .finally , they also visit significantly less search and porn websites than average tor users. this might be because they already rely on bittorrent rather than the web to search and download content , including pornographic material .tor s efficiency has required several adaptations to the original design of onion routing whose impact on privacy are not well understood in practice . in particular , we showed that the multiplexing of several streams into a single circuit can significantly degrade privacy .to date , tor measurement studies and attacks have been carried out in isolation .measurement studies have documented _ who is using tor _ and _ how tor is used _ but without the ability of associating the two information , e.g. , to profile tor users .attacks have documented methodologies to associate the two information but without actually profiling tor users .this paper strikes a balance between the two by developing new attacks targeted to p2p applications , launching these attacks at a reasonable scale against the tor network , and profiling tor users .   for simplicity ,we assume that the attacker always controls an exit node in the attacks described hereafter .we now describe attacks targeting web applications to reveal the ip address of tor users .fortconsult designed two attacks based on active content injection by the exit nodes to trace tor users . the first attacks consisted in flash injection so that the targeted user connects to a server controlled by the attacker _ outside _ of tor , hence exposing his ip address .( a cookie is used to associate that ip address to the stream in tor . ) the second attack consisted in injecting javascript to send the ( local ) ip address of the user over tor .this study reported that whereas the first attack was effective , the second was not mainly because local ip addresses sent over tor were not routable , e.g. , 192.168.0.1 .abbott et al .relied on javascript and html meta refresh tag to inject timing patterns .the assumption being that users will leave that page open long enough so that pattern can be spotted by an entry node , also controlled by the attacker ( thus tracing the user ) .it is interesting to note that even users having disabled active content are susceptible to html meta refresh tag injection .  to the best of our knowledge ,we are the first to design attacks against p2p applications on tor , to validate these attacks at a reasonable scale , and to demonstrate that one can associate many streams possibly originating from `` secure '' ( i.e. , proxied ) browsers , with traced users .we remark that even though we have targeted p2p applications in this paper , the bad apple attack can originate from any insecure application .   the main measurement study of tor that we are aware of has been made by mccoy et al . . the authors provided interesting insights into_ who is using tor _ and _ how tor is used and mis - used_. in particular , they showed that bittorrent generates 40% of all traffic on tor and claimed that it represents only 3% of all streams .we complemented this measurement study in two important aspects .first , we showed that 70% of bittorrent users establish p2p connections _ outside _ of tor thus making most bittorrent streams ( and traffic ) invisible to the tor network .we argue that this finding makes of bittorrent users a target of choice on tor .second , we launched attacks to profile tor users .this profiling brought elements of answer to one important question raised by mccoy et al ., that is , `` ... why there is such a large scale adoption of tor in [ ... ] specific countries , relative to tor usage in other countries .'' we showed that the answer to that question ( at least for bittorrent ) is unlikely to be technological or political but in fact sociological .using bittorrent as an insecure application , we designed two attacks , one consisting in hijacking tracker responses and one exploiting the statistical properties of the dht , to trace bittorrent streams on tor .we then showed that the bad apple attack allows us to trace non - bittorrent streams .in particular , we traced 193% of additional streams as compared to bittorrent streams , including 27% of http streams possibly originating from `` secure '' (i.e. , proxied ) browsers . in total, we traced 9% of _ all _ tor streams carried by our instrumented exit nodes .we ran these attacks in the wild for 23 days and reveal 10,000 ip addresses of tor users . using these ip addresses ,we then profiled not only the bittorrent downloads but also the websites visited per country of origin of tor users . in particular, we showed that sociological reasons could explain the large number of tor users in certain countries relatively to other .finally , we presented results that suggest the existence of an underground ecosystem on tor .defending against the bad apple attack is not straightforward .the most effective defense would be to have one stream per circuit , as in the original onion routing , however , performances issues make this defense unfeasible. another defense would be to isolate streams by groups of destination port in different circuits , e.g. , the secure and the insecure circuit .destination ports known to be used by secure applications , e.g. , 80 , 22 , would use the secure circuit thus limiting the risk that the source ip address of one stream in that circuit gets revealed by an insecure application .one weakness of this defense is that an attacker could trick an insecure application into connecting to a port that is usually used by a secure application , thus multiplexing the insecure stream into the secure circuit .yet another defense would be to isolate each application into its own circuit , hence compartmenting the bad apple attack to the insecure application .however , modern operating systems lack a portable way to map an incoming stream to an application .we have discussed our results and possible solutions to address the bad apple attack with the tor project .we remark that even though the bad apple attack does not exist in application - level anonymity networks dedicated to a single application ( e.g. , oneswarm ) , the corpus of networking applications is too broad to practically build one network for each application . in this respect , we believe that we have validated an important attack against the design of modern anonymity networks and that we should defend against it to protect users privacy on the internet .* acknowledgements * we warmly thank keith ross and chao zhang for sharing their identifiers of public and private content with us .we are also grateful to roger dingledine , nick mathewson , the anonymous reviewers , and our shepherd , thorsten holz , for their constructive comments .

Abstract: tor is a popular low - latency anonymity network . however , tor does not protect against the exploitation of an insecure application to reveal the ip address of , or trace , a tcp stream . in addition , because of the linkability of tor streams sent together over a single circuit , tracing one stream sent over a circuit traces them all . surprisingly , it is unknown whether this linkability allows in practice to trace a significant number of streams originating from secure ( i.e. , proxied ) applications .  in this paper , we show that linkability allows us to trace 193% of additional streams , including 27% of http streams possibly originating from `` secure '' browsers . in particular , we traced 9% of _ all _ tor streams carried by our instrumented exit nodes . using bittorrent as the insecure application , we design two attacks tracing bittorrent users on tor . we run these attacks in the wild for 23 days and reveal 10,000 ip addresses of tor users . using these ip addresses , we then profile not only the bittorrent downloads but also the websites visited _ per country of origin of tor users_. we show that bittorrent users on tor are over - represented in some countries as compared to bittorrent users outside of tor . by analyzing the type of content downloaded , we then explain the observed behaviors by the higher concentration of pornographic content downloaded at the scale of a country . finally , we present results suggesting the existence of an underground bittorrent ecosystem on tor .


polynomial - in - time regression is one of the longest - established tools of time series analysis ( see jones  ) . in much empirical work ,especially when stochastic trends , such as unit roots , are also involved , only a linear trend is countenanced , or merely a constant intercept . on the other hand ,classical methods can test polynomial order when observations are equally spaced in time . with independent and identically distributed ( i.i.d . )normal errors , a particularly elegant way of achieving this , with finite sample validity , results from an orthogonal polynomial representation  the covariance matrix of the least - squares estimate ( lse ) is diagonalized , and contributions to the  statistic from individual regressors are i.i.d . ( see section 3.2.2 of anderson ) .asymptotic theory is valid under much wider conditions on the errors ; indeed from section 7.4 of grenander and rosenblatt , the lse is asymptotically efficient ( in the gauss markov sense ) when the ( possibly non - gaussian ) errors are covariance stationary with spectral density bounded and bounded away from zero at zero frequency , as with short memory processes .polynomial models have also been extended to spatial lattice data ( see section 3.4 of cressie ) .polynomials are nevertheless restrictive .the weierstrass theorem justifies their uniform approximation of any continuous function over a compact interval , but seems less practically relevant the longer the data set .nonparametric smoothing may be unreliable in a series of moderate length , when instead richer parametric models than polynomials might be considered .one class that advantageously nests polynomials , which has received little theoretical attention , consists of `` generalized polynomial '' or `` power law '' models .with equally spaced time series observations , , consider where the and are real valued and all can be unknown , for all , and the zero - mean unobservable process is covariance stationary with short memory . for , would not be estimable ( whether were known or unknown ) because the corresponding signal is drowned by the noise . for , is estimable but we omit this possibility because our central limit theorem requires to lie in the interior of a compact set .polynomials , such as when for all , are nested ; indeed this is a hypothesis that might be tested within ( [ eq1.1 ] ) .we consider the nonlinear least - squares estimate ( nlse ) of the in ( [ eq1.1 ] ) and , more generally , of exponents and coefficients in an extended model defined on a lattice , applying to spatial and spatio - temporal data , where our provision , for example , for weaker trends than linear ones and for decaying trends seems practically useful . unlike the lse when exponents are known, the nlse can not be expressed in closed form and requires numerical optimization .correspondingly , asymptotic theory , with sample size  increasing , is needed to justify rules of statistical inference even when errors are gaussian .we establish consistency and asymptotic normality for the nlse of exponent and coefficient estimates , achieving also an analogous efficiency bound to that described above . as with other implicitly defined estimates, asymptotic distribution theory makes use ( in application of the mean value theorem ) of an initial consistency proof .many such proofs ( see jennrich  , malinvaud ) require regressors to be non - trending , whence under suitable additional conditions all parameter estimates are -consistent . for the nlse of ( [ eq1.1 ] ) , wu significantly relaxed this requirement but nevertheless appears to heavily restrict the diversity of trends .the discussion after assumptions a and a of wu indicates that they reduce in ( [ eq1.1 ] ) with known to the assumption , and no weaker requirement suffices in the case of unknown  .example 4 of wu addressed the latter case but with only ( and for  ] , , .we varied absolutely and also the relative  , taking , ..3lld2.3d2.3l@ & & & & & & & & + & & bias & 0.008&0.007&0.024&0.000&0.017&0.000 + & & mse & 0.016&0.007&0.080&0.001&0.051&0.000 + & & size5 & 0.100&0.125&0.151&0.048&0.166&0.055 + & & size1 & 0.044&0.048&0.075&0.010&0.084&0.010 + & & bias & 0.005&0.009&0.016&-0.001&0.009&0.002 + & & mse & 0.010&0.009&0.060&0.006&0.063&0.007 + & & size5&0.132 & 0.132&0.180&0.053&0.186&0.051 + & & size1&0.055&0.050&0.084&0.015&0.090&0.011 + & & bias & -0.002&0.002&0.016&0.000&-0.007&0.000 + & & mse & 0.003&0.001&0.022&0.000&0.010 & 0.000 + & & size5 & 0.086&0.104&0.115&0.039&0.120&0.051 + & & size1&0.030&0.039&0.051&0.005&0.049&0.012 + & & bias & 0.003&0.002&0.006&0.000&-0.001&0.000 + & & mse & 0.002&0.002&0.013&0.000&0.013&0.000 + & & size5 & 0.074&0.075&0.108&0.043&0.103&0.039 + & & size1&0.024&0.022&0.033&0.010&0.037&0.010 +  our first experiment took the to be i.i.d . variables .tables [ tab1 ] and [ tab2 ] report , for the respective parameter combinations , bias ( bias ) , mean squared error ( mse ) , and empirical size at 5% ( size5 ) and 1% ( size1 ) for the nlse , , and also , the lse of that correctly assumes ,  , across replications .the sizes were proportions of significant estimates , using normal critical values scaled by estimated standard deviations which , in the case of the , , were computed on the basis of theorem [ th3 ] with current parameter estimates replacing true values of , and replaced by the sum of squared residuals divided by ( so the spatial independence of the was treated as known , as it was also in the conventional scaling used for the ) .the tables reveal a definite inferiority of the nlse relative to the lse , but unsurprisingly , as the lse is exactly unbiased , more efficient and yields exact critical regions .though the nlse - based tests on are nearly always over - sized , this phenomenon diminishes with increased , and overall the discrepancy between the performances of the two classes of the estimate does not seem very serious .there is also a predominate over - sizing of the tests on , but again this falls as increases , and , in table [ tab2 ] in particular , it is often modest .there is a tendency for the nlse to over - estimate , but for biases only exceed 2% of the parameter value when and .for they never reach 1% , while overall they mostly fall with increasing , as does the mse . in table[ tab2 ] , the results are not in line with what the rates in theorem [ th3 ] suggest , because the fall in mse is greater for and than for and , despite the fact that and .nevertheless , it is not clear to what extent one would expect asymptotic theory to predict comparisons at this level of refinement in such sample sizes .note that the monte carlo results are also difficult to judge relative to the theory because the various  did not result from fixing the and and then increasing , but were chosen with a view to representing some variability in , and relative to and .in addition , the convergence rates of and do not only depend on , but on the overall .other results are more closely in line with the asymptotic theory .this is the case in table [ tab1 ] where , with , the above mse ratios are sometimes greater for and/or  and sometimes less .it is also the case in table [ tab2 ] for the lse , as elsewhere , that comparisons are sometimes difficult as a number of mses are zero to , and even to ( unreported here ) , decimal places ..3d2.3d2.3d2.3d2.3@ & & & & & & & & + & & bias & 0.008&0.001&0.024&0.003&-0.002&-0.000 + & & mse & 0.014&0.001&0.071&0.005&0.001&0.000 + & & size5 & 0.063&0.060&0.087&0.077&0.053&0.090 + & & size1&0.028&0.012&0.038&0.029&0.014&0.034 + & & bias & 0.008&0.000&0.020&0.004&0.000&-0.000 + & & mse & 0.013&0.003&0.074&0.004&0.001&0.000 + & & size5&0.069&0.057 & 0.101&0.058&0.065&0.039 + & & size1&0.033&0.013&0.047&0.015&0.017&0.009 + & & bias & 0.005&-0.000&-0.001&-0.002&0.000&0.000 + & & mse & 0.005&0.000&0.028&0.002&0.000 & 0.000 + & & size5 & 0.052&0.054&0.069&0.030&0.059&0.041 + & & size1&0.017&0.012&0.017&0.012&0.011&0.006 + & & bias&0.002 & 0.001&0.004&0.001&0.004&0.000 + & & mse & 0.004&0.001&0.025&0.001&0.000&0.000 + & & size5 & 0.058&0.044&0.070&0.081&0.043 & 0.055 + & & size1&0.018&0.011&0.019&0.019&0.010&0.020 +  next we considered the effect of dependence , employing three different models for , again with .all models entailed weak dependence , with varying spans , but in the first dependence was negative , so that the spectral density at zero was small , whereas in the other two it was positive , producing a peaked spectral density . in the following , i.i.d . .  1 .  multiple direction : 2 .multilateral , no interactions : for , , , .3 .  bilateral , on diagonal :   .3d2.4d2.3d2.3ld2.3@ & & & & & & & & + & & bias & 0.005&0.003&0.006&0.000&0.002&-0.000 + & & mse&0.006&0.003&0.031&0.000&0.021&0.000 + & & bias&0.005 & 0.001&0.001&0.000&0.008&-0.000 + & & mse&0.003&0.003&0.023&0.000&0.023&0.000 + & & bias&0.001 & 0.0001&0.001&-0.000&0.001&0.000 + & & mse&0.001&0.000&0.006&0.000&0.003&0.000 + & & bias & 0.002&-0.001&-0.003&-0.000&0.005&0.000 + & & mse&0.000&0.000&0.004&0.000&0.004&0.000 +  .3lld2.3d2.3d2.3@ & & & & & & & & + & & bias & 0.003&0.000&0.003&-0.000&-0.000&0.000 + & & mse&0.003&0.000&0.017&0.001&0.000&0.000 + & & bias & -0.003&0.000&0.014&-0.001&-0.001&0.000 + & & mse&0.003&0.000&0.018&0.001&0.000&0.000 + & & bias & -0.000&0.000&0.003&0.000&-0.000 & -0.000 + & & mse&0.001&0.000&0.004&0.000&0.000&0.000 + & & bias & -0.001&0.000&0.005&0.001&-0.000&-0.000 + & & mse&0.000&0.000&0.004&0.000&0.000&0.000 +  for the same parameter values as before , bias and mse of the lse and nlse are presented in tables [ tab3][tab8 ] , with tables [ tab3 ] and [ tab4 ] referring to ( [ eq3.1 ] ) , tables [ tab5 ] and [ tab6 ] to ( [ eq3.2 ] ) , and tables [ tab7 ] and [ tab8 ] to ( [ eq3.3 ] ) . as before the lse are exactly unbiased , as the monte carlo results tend to illustrate .however , perhaps surprisingly , the dependent model ( [ eq3.3 ] ) produces some very large biases in the nlse , though not so much in . for the other dependence models the nlse biases are not necessarily greater than under independence .the mse magnitudes are not directly comparable to those of tables [ tab1 ] and [ tab2 ] , because scales were not calibrated , but a similar overall picture emerges : the nlse of often has much greater mse than the lse , but this falls with increasing , as does that of the nlse of . in tables[ tab4 ] , [ tab6 ] and [ tab8 ] , where , , the same somewhat surprising feature as noted in table [ tab2 ] appears , with and improving less than and with increasing , and the only additional point to add to our previous discussion is that convergence is often expected to be slowed by dependence ..3ll@ & & & & & & & & + & & bias & 0.032&0.020&0.053&-0.001&0.035&0.000 + & & mse&0.050&0.026&0.249&0.004&0.169&0.002 + & & bias & 0.029&0.020&0.017&-0.005&0.047&0.003 + & & mse&0.031&0.031&0.181&0.003&0.177&0.003 + & & bias & 0.010&0.003&0.017&-0.001&0.015&0.001 + & & mse&0.013&0.004&0.091&0.001&0.045&0.000 + & & bias & 0.008&0.007&0.006&-0.001&0.014&0.000 + & & mse&0.007&0.008&0.059&0.000&0.060&0.001 +  .3ld2.3d2.3d2.3@ & & & & & & & & + & & bias & 0.064&0.001&0.048&0.005&-0.001&-0.000 + & & mse&0.115&0.000&0.272&0.024&0.003&0.000 + & & bias & 0.067&-0.001&0.023&-0.002&0.005&0.000 + & & mse&0.111&0.001&0.267&0.019&0.005&0.000 + & & bias & 0.019&0.000&0.035&0.000&-0.001&0.000 + & & mse&0.027&0.000&0.151&0.009&0.000&0.000 + & & bias & 0.008&0.000&0.046&-0.002&-0.001&0.000 + & & mse&0.020&0.000&0.143&0.007&0.001&0.000 +for known , long - established techniques ( see , section 2.6 ) give ( where is defined near the start of appendix below ) , so ignorance of  incurs not only efficiency loss , but slightly slower convergence .theorem [ th3 ] also implies a singularity in the limit distribution , whose covariance matrix has rank only .this is due to bias in , which on expansion is seen to have a term linear in that dominates the contribution from .nevertheless , theorem [ th3 ] does provide separate inference on ( moreover , one can conduct joint inference that does not cover both and for any ) , though , given assumption [ ass1 ] , we can not test zero restrictions on  . in our setting , may be of less initial interest than , and theorem [ th3 ] allows inference on  with converging slightly faster than , and at what appears to be the optimal rate for this problem .if independence of the is not assumed , the limiting covariance matrix in theorem [ th3 ] can be consistently estimated ( under additional conditions ) by replacing by a parametric or smoothed nonparametric estimate based on nlse residuals ..4ld2.3@ & & & & & & & & + & & bias & 0.074&0.096&0.154&0.008&0.091&-0.004 + & & mse&0.129&0.157&0.738&0.048&0.549&0.024 + & & bias & 0.041&0.069&0.105&-0.008&0.050&0.008 + & & mse&0.080&0.097&0.455&0.033&0.371&0.032 + & & bias & 0.016&0.036&0.134&0.0010&0.017&-0.000 + & & mse&0.043&0.032&0.462&0.014&0.232&0.005 + & & bias & 0.013&0.024&0.061&-0.003&0.028&0.002 + & & mse&0.026&0.026&0.214&0.009&0.182&0.009 +  .3d2.3d2.3d2.3ld2.3@ & & & & & & & & + & & bias & 0.063&-0.000&0.100&0.014&0.009&-0.000 + & & mse&0.518&0.003&1.217&0.291&0.019&0.000 + & & bias & 0.098&-0.000&0.118&0.009&0.008&-0.000 + & & mse&0.512&0.003&0.912&0.222&0.016&0.000 + & & bias & -0.037&-0.002&-0.007&-0.001&0.008&0.000 + & & mse&0.275&0.000&1.059&0.128&0.004&0.000 + & & bias & 0.054&0.000&0.128&-0.001&0.001 & 0.000 + & & mse&0.226&0.000&0.616&0.086&0.003&0.000 +  the form of the limiting covariance matrix in theorem [ th3 ] , with dependence simply reflected in the scale factor , suggests that a generalized nlse , which corrects parametrically or nonparametrically for correlation in , affords no efficiency improvement ( cf .section 7.4 of grenander and rosenblatt ) .on the other hand , our estimates are not fisher efficient for non - gaussian .departures from gaussianity might be detected by , for example , nonparametric probability density estimation based on nlse residuals ; hallin , lu and tran studied density estimation for linear lattice processes .more efficient parameter estimates could be obtained by -estimation using a correctly parameterized distribution , or adapting semi - parametrically to a nonparametric one , in either case employing parametric or approximating them via a long autoregression .the extra proof details would be far from trivial , but convergence rates should be unaffected , with the limiting covariance matrix of theorem [ th3 ] simply shrunk by a scalar factor .another extension allows long or negative memory , in , bearing in mind results of yajima for ( [ eq1.1 ] ) with known integer , and yajima and matsuda ; this would affect all convergence rates by the same scalar factor , the efficiency property in comment 3 would be lost , and negative , and corresponding may not be estimable .  in an alternative formulation to ( [ eq1.1 ] ) , is replaced by , confining the regression to the unit interval , and ( [ eq2.1 ] ) can be analogously modified .consistency is then much easier to prove , all exponent estimates being -consistent .a similar device is employed in fixed - design nonparametric regression , but unlike there it is not essential in order to achieve consistency in our parametric setting , where we find it aesthetically unattractive given that is defined on an increasing domain .the results are straightforwardly extended to allow some in ( [ eq2.1 ] ) to be known ; for example , to specify an intercept by , though the norming factor and limit covariance matrix in theorem [ th3 ] are affected .our notation suggests constant spacing between observations across all dimensions , but allowing the interval of observation to vary with dimension affects each by a factor depending also on the corresponding , but not the themselves .irregular spacing of observations , either due to missing data from an otherwise regular lattice , or with observations occurring anywhere on , can also be considered . in both of these settingsasymptotic theory requires a degree of regularity in the observation locations , ruling out situations where observations become too sparse , for example .given this , the extension is relatively simple with independent . under dependence , asymptotic variance formulae will be complicated by the irregular spacing and the efficiency property of comment 3 will be lost .in addition , different kinds of assumptions from ours on the errors may be needed . in the case of missing data from an otherwise regular lattice , our assumptions [ ass3 ] ( for consistency ) and [ ass5 ] ( for asymptotic normality ) should still suffice .but for observations anywhere on it would be appropriate to consider an underlying continuous process . then , for consistency , a suitable ergodicity property would be needed , whereas for asymptotic normality leading possibilities that can entail weak dependence analogous to that of assumption [ ass5 ] include suitable linear functionals of brownian motion and mixing conditions .a bayesian treatment would be worthwhile , with suitable priors placed on the exponents and possibly also the coefficients .when a more realistic model than ( [ eq2.1 ] ) might allow interaction terms , that is , products of powers of and , .our proof methods are extendable , but from a practical perspective the curse of dimensionality threatens and the issue of parsimonious specification , already posed by ( [ eq2.1 ] ) , becomes more pressing. a penalized procedure could be used .modified model classes might provide an alternative route to parsimony ; for example , one might take with replaced by for known or unknown ( cf .example 3 of wu ) .trigonometric factors might also be incorporated ( cf . section 7.5 of grenander and rosenblatt ) .  for alternative classes of trending model ( for example , involving wavelets ) , asymptotic estimation theory might be handled by similar techniques .an alternative practically relevant modelling of the treats them as heteroscedastic but possibly independent .broadly similar proof techniques would provide corresponding results to ours , but the nlse is less efficient than a suitably weighted estimate.=1  though we have focussed on ( [ eq1.1 ] ) and ( [ eq2.1 ] ) to fix ideas , our methods and theory can be developed to cover models that incorporate power law trends along with other explanatory variables , both stochastic and non - stochastic , such as extensions of the nonparametric and semiparametric spatial regressions considered by gao , lu and tjstheim  and lu , lundervold , tjstheim and yao , and so the paper can be viewed as introducing machinery relevant to a wide variety of settings.=-1we present a consistency theorem for a general , implicitly defined extremum estimate under unprimitive conditions that will be checked in the paper s setting and seem capable of checking in a number of others .as this appendix is self - contained , there seems no risk of confusion in employing notations that are similar to those elsewhere in the paper but can have slightly different meanings .we estimate the vector parameter , with elements  , , by , where depends on sample size and  is a fixed compact set . for positive scalars , , depending on and such that , , define , and \\[-10pt ] \nonumber \bar{\mathcal{n}}(c_{w } ) &= & \theta\setminus\mathcal { n}(c_{w}),\qquad \mathcal{s}_{w}=\bar{\mathcal{n}}(c_{w})\cap\mathcal{n}(c_{w+1}).\end{aligned}\ ] ]  [ tha ] assume :   for a finite integer and sufficiently large ;  there exist positive and , such that and , and as , and \label{eqa.3 }  \sup_{h\in\mathcal{s}_{w}}\frac{\vert v(h ) \vert}{s_{w}}&=&\mathrm{o}_{p}(1).\end{aligned}\ ] ] then where is a vector with element .we show that as . by a standard kind of argument under ( i ) , . thus the last probability is bounded by which is bounded by which tends to zero on applying ( [ eqa.2 ] ) and ( [ eqa.3 ] ) .three comments are relevant .( 1 ) in the setting of the rest of the paper , can be chosen nonstochastic but this is not possible in the context of such stochastic trends as unit roots , where the more general ( [ eqa.2 ] ) is useful .( 2 ) an almost sure convergence version of theorem [ tha ] is possible under suitably strengthened versions of ( [ eqa.2 ] ) and ( [ eqa.3 ] ) . ( 3 ) by comparison with our decomposition of into , van de geer ( see pages 69 , 70 ) employed a `` peeling device '' to obtain an exponential inequality for , where is a stochastic process , is a non - negative function and the set is `` peeled off '' as , where , for an increasing sequence and need not be finite .thus and only the supremum of the numerator of the original statistic need be approximated .there is no denominator there like in our problem , and our decomposition of is designed to suitably balance and on each  to enable choices of the that make all summands in ( [ eqa.4 ] ) small .to define , introduce first , for , the vector with element and the matrix with element for , , where , for all . for , , introduce the matrix with  block when and when .denote .writing , , define matrices , with  block , when and , when , where `` '' denotes the hadamard product .put .define where is the diagonal matrix such that and is the vector of s .proof of theorem [ th1 ] we have  , , where for , .the subsequent proof implies that after suitable norming is well conditioned for relevant and large  . in theorem[ tha ] , take , for , , , with , .define , for , , and a finite , positive scalars , , such that for each such .define define neighbourhoods , , , . finally , define for , and then , as in ( [ eqa.1 ] ) .take , , , so we need to show that as . we check ( i ) and ( ii ) of theorem [ tha ] , where ( [ eqa.2 ] ) reduces to the requirement for large enough and as in ( [ eqa.2 ] ) . from ( [ eqb.1 ] ) and ( [ eqb.2 ] ) , where it follows from proposition [ pr1 ] that thus ( [ eqa.2 ] ) is satisfied when next , ( [ eqa.3 ] ) is implied if  \sup_{h\in s_{w}}\vert v_{2}(h)-v_{2}(\theta ) \vert & = & \mathrm{o}_{p}(s_{w } ) , \label{eqb.5 } \\[-2pt ]\sup_{h\in s_{w}}\vert v_{3}(h)\vert&=&\mathrm{o}_{p}(s_{w } ) , \label{eqb.6}\vspace*{-2pt}\end{aligned}\ ] ] as .note that in ( [ eqb.5 ] ) we are considering the difference for suitably close to and this closeness is important in obtaining the desired result , whereas in the usual kind of consistency proof , for standard , non - mixed rate settings , one more simply shows the convergence to zero in probability of a suitably normalized , uniformly in  .now ( [ eqb.6 ] ) follows from proposition [ pr4 ] , while ( [ eqb.4 ] ) and ( [ eqb.5 ] ) follow from propositions [ pr2 ] and [ pr3 ] , respectively , if where  ] with  and , with  q_{2}^{(2)}(a ) & = & 2\sum _ { u\in\mathbb{n}}\ { b^{\prime } f(u;h)-\beta^{\prime}f(u;\theta)\ } j(u;h , b),\end{aligned}\ ] ] in which is the 2 matrix with  block for , for , and for , , being the diagonal matrices such that , .  by the mean value theorem where is formed from by evaluating its row at , where , . by proposition [ pr5 ] ( [ eqb.9 ] )is let and be the matrix with blocks  , , with . noting proposition [ pr6 ] and the representations b_{\delta}\gamma b_{\delta}d_{+}^{-1}l_{+}q^{(1)}(\alpha ) & = & -2n^{- { 1/2}}\sum _ { u\in\mathbb{n}}\bigl [ ( \beta_{\delta } ^{-1}\lambda ^{\prime } ) ^{\prime},\bigl ( l^{-1}\lambda\ { l(u)-l \ } -\lambda^{\prime}\bigr ) ^{\prime}\bigr ] ^{\prime } \\[-2pt ] & & \hspace*{53pt}{}\times d^{-1}f(u;\theta)x_{u},\end{aligned}\ ] ] we obtain from ( [ eqb.9 ] ) d^{-1}f(u;\theta)x_{u } \\[-2pt ] & & { } -n^{- { 1/2 } } \sum _ { u\in\mathbb{n}}\bigl [ 0,\bigl ( l^{-1}\lambda \{l(u)-l\}d^{-1}f(u;\theta)x_{u}\bigr ) ^{\prime}\bigr ] ^{\prime } \\[-2pt ] & & { } + \mathrm{o}_{p } ( ( \log n)^{-2 } ) n^{-{1/2 } } \sum _ { u\in\mathbb{n } } ( ( \beta_{\delta}l(u ) ) ^{\prime},l ) ^{\prime}d^{-1}f(u;\theta)x_{u}.\end{aligned}\ ] ] the last two terms are by application of lemmas [ le15 ] and [ le10 ] , respectively .the proof is completed by applying proposition [ pr7 ] to the first term .[ pr1 ] for all given by such that , , , there exists such that , for all   non - singularity of for , and where throughout denotes a finite , positive generic constant , follow from lemmas [ le2 ] and [ le3 ] , numerators of elements of the inverse being bounded and denominators bounded away from zero .now ^{-1} ] , .[ le6 ] for   the integral is   [ le7]for , }\biggl\ { ( a - b)^{-2}\sum_{j=1}^{j}v\biggl ( \frac { j}{j};a , b\biggr ) ^{2}\biggr\ } \leq kj(\log j)^{2}.\ ] ]  by the mean value theorem , ,\ ] ] where . also , for such , .\ ] ]thus the quantity in braces in ( [ eqd.9 ] ) is bounded by because .[ le8]for , }\vert a - b\vert^{-1}\biggl\vert\frac{1}{j}\sum_{j=1}^{j}v\biggl ( \frac{j}{j};a , b\biggr ) -\int _ { 0}^{1}v ( x;a , b ) \,\mathrm{d}x\biggr\vert\leq\frac{k(\log j)^{2}}{j^{1+\min(\underaccent\bar { a},0)}}.\ ] ]  the expression within the modulus is from ( [ eqd.10 ] ) and ( [ eqd.11 ] ) , the last integral is bounded by and the same bound results for the penultimate term of ( [ eqd.13 ] ) . by the mean value theorem bounded by where , and the first modulus is bounded by thus the first term of ( [ eqd.13 ] ) is bounded by times  [ le9]for , }}}_{i=1,2}}\biggl\ { \prod^{2}_{i=1}\vert a_{i}-b_{i}\vert\biggr\ } ^{-1}\biggl\vert\frac{1}{j}\sum_{j=1}^{j } \prod^{2}_{i=1}v\biggl ( \frac { j}{j};a_{i},b_{i}\biggr )  -\int _ { 0}^{1}\prod^{2}_{i=1 } v(x;a_{i},b_{i})\,\mathrm{d}x\biggr\vert\\ & & \quad\leq\frac{k(\log j)^{3}}{j^{1+\min(2\underaccent\bar{a},0)}}.\end{aligned}\ ] ]  the expression within the second modulus is \\[-8pt ] \nonumber & & \quad{}+j^{-1}\prod^{2}_{i=1}v\biggl ( \frac{1}{j};a_{i},b_{i}\biggr ) -\int _ { 0}^{1/j}\prod^{2}_{i=1 } v(x;a_{i},b_{i})\,\mathrm{d}x.\end{aligned}\ ] ] similarly to the proof of lemma [ le7 ] , the last term is bounded by the expression in braces in ( [ eqd.17 ] ) can be written both terms are treated similarly ; we consider only the first . from the bounds ( [ eqd.14 ] ) , ( [ eqd.15 ] ) its first factoris bounded by , and its second one by .thus its contribution is whence the result follows by an analogous calculation to ( [ eqd.16 ] ) .[ le10]for and _ _ , and all _ _ } \biggl\vert n^{- { 1/2 } } \sum_{u\in\mathbb{n}}\biggl ( \frac{u_{i}}{n_{i}}\biggr ) ^{a}(\log u_{i})^{q}x_{u}\biggr\vert\biggr\ } \leq k(\log n)^{q}.\ ] ]  by summation by parts where is with replaced by . thus the expression in the modulus in ( [ eqd.18 ] ) is where the factor in braces in ( [ eqd.19 ] ) is bounded by , whereas .thus the left side of ( [ eqd.18 ] ) is bounded by since and & \leq&\frac{kns}{n_{i}}(\log s)^{2q } \sum_{u\in\mathbb{z}^{d}}\vert\gamma_{u}\vert\leq\frac{kns(\log s)^{2q}}{n_{i}}.\vspace*{-5pt}\end{aligned}\ ] ]  [ le11 ] for , } \vert a - b\vert^{-1}\biggl\vert\sum_{u\in\mathbb{n } } v ( u_{i}/n_{i};a , b ) x_{u}\biggr\vert\biggr\ } \leq kn^{{1/2}}\log n.\vspace*{-2pt}\ ] ]  by summation by parts , from ( [ eqd.14 ] ) and ( [ eqd.15 ] ) , the expression in braces is bounded by thus the left side of ( [ eqd.20 ] ) is bounded by , which , from the proof of lemma [ le10 ] ( with ) , has the desired bound .[ le12 ] let be a scalar and be a sequence such that as , for some .then for all ,   the left side is bounded by  & \leq&kj^{\eta/2}\mathrm{o}_{p } ( j^{-\eta } ) \frac{1}{j } \sum_{j=1}^{j}\biggl ( \frac{j}{j}\biggr ) ^{a}=\mathrm{o}_{p } ( j^{-\eta /2 } ) .\vspace*{-6pt}\end{aligned}\ ] ]  [ le13]for , there is an such that for all sufficiently large ,   the left side is bounded by the first modulus is bounded by for $ ] , .thus the first term of ( [ eqd.23 ] ) is for , for , and for .the last integral is . since there is an to satisfy ( [ eqd.21 ] ) .[ le14 ] for any , there is an such that for all sufficiently large ,   the left side is bounded by the first integrand is bounded by as in the proof of lemma [ le13 ] ; the proof is completed in similar fashion .  [ le15 ] for any and all sufficiently large ,   the left side is by assumption [ ass3 ] and straightforward application of lemmas [ le13 ] and [ le14 ] .[ le16]for , , and any finite positive or negative integer , there is an such that for all sufficiently large , \\[-8pt ] \nonumber & & \quad \leq\vert m\vert j^{-\eta}.\end{aligned}\ ] ]  we have by elementary inequalities the left side of ( [ eqd.23 ] ) is bounded by which is thank the editor , associate editor and two referees for helpful comments .i am also grateful to francesca rossi for carrying out the monte carlo simulations .this research was supported by esrc grant res-062 - 23 - 0036 , spanish plan nacional de i+d+i grant sej2007 - 62908/econ , and a catedra de excelencia at universidad carlos iii , madrid .

Abstract: power law or generalized polynomial regressions with unknown real - valued exponents and coefficients , and weakly dependent errors , are considered for observations over time , space or space time . consistency and asymptotic normality of nonlinear least - squares estimates of the parameters are established . the joint limit distribution is singular , but can be used as a basis for inference on either exponents or coefficients . we discuss issues of implementation , efficiency , potential for improved estimation and possibilities of extension to more general or alternative trending models to allow for irregularly spaced data or heteroscedastic errors ; though it focusses on a particular model to fix ideas , the paper can be viewed as offering machinery useful in developing inference for a variety of models in which power law trends are a component . indeed , the paper also makes a contribution that is potentially relevant to many other statistical models : our problem is one of many in which consistency of a vector of parameter estimates ( which converge at different rates ) can not be established by the usual techniques for coping with implicitly - defined extremum estimates , but requires a more delicate treatment ; we present a generic consistency result .