On the Functional Empirical Process and Its Application to the Mutual Influence of the Theil-Like Inequality Measure and the Growth

We set in this paper a coherent theory based on functional empirical processes that allows to consider both the poverty and the inequality indices in one Gaussian field in which the study of the influence of the one over the other is done. We use the General Poverty Index (GPI), that is a class of poverty indices gathering the most common ones and a functional class of inequality measures including the Entropy Measure, the Mean Logarithmic Deviation, the different inequality measures of Atkinson, Champernowne, Kolm and Theil called Theil-Like Inequality Measures (TLIM). Our results are given in a unified approach with respect to the two classes instead of their particular elements. We provide the asymptotic laws of the variations of each class over two given periods and the ratio of the variation and derive confidence intervals for them. Although the variances may seem somehow complicated, we provide R codes for their computations and apply the results for the pseudo-panel data for Senegal with a simple analysis.


Introduction
In many cases, one has to monitor a specific situation through some risk measure J on some population. The variation of J over time is called growth in case of negative variation and recession alternatively. This growth or recession is not itself sufficient to describe the improvement or deterioration of the situation. Often, the distribution of the underlying variable over the population should also be taken into account in order to check whether the growth concerns a great number of individuals or is rather concentrated on a few numbers of them.
In the particular case of welfare analysis, one may measure poverty (or richness) with the help of poverty indices J based on the income variable X. Over two periods s = 1 and t = 2, we say that we have a gain against poverty when , or simply a growth against poverty. Before claiming any victory, one must be sure that, meanwhile, the income did not become more unequally distributed, that is the appropriate ine-quality coefficient I did not increase. One can achieve this by studying the ratio To make the ideas more precise, let us suppose that we are monitoring the poverty scene on some population over the period time [1,2] and let  1 2 ,  X X be the income variable of that population at periods 1 and 2. Let us consider one sample of individuals or households, and observe the income couple 1 n    and By particularizing the functions A and w and by giving fixed values to the   0,1 .  x s  we may find almost all the available indices, as we will do it later on. In the sequel, (1) will be called a poverty index (indices in the plural) or simply a poverty measure according to the economists' terminalogy.
This class includes the most popular indices such as those of Sen [4], Kakwani [5], Shorrocks [6], Clark-Hemming-Ulph [7], Foster-Greer-Thorbecke [8], etc. See Lo [2] for a review of the GPI. From the works of many authors ( [9,10] for instance),   n J i is an asymptotically sufficient estimate of the exact poverty measure As for the inequality measure, we use this Theil-like family, where we gathered the Generalized Entropy Measure, the Mean Logarithmic Deviation [11][12][13], the different inequality measures of Atkinson [14], Champernowne [15] and Kolm [16] in the following form: X that we suppose to be finite here.
Each measure of the Theil-like family has its own particular properties, derived from the combination of different concepts. One may mention the concept of welfare criteria (Atkinson [14], Sen [17]), that of the analogy with analysis of risks (Harsanyi [18,19]; Rothschild and Stiglitz [20]), the complaints approach (Temkin [21]) etc. The Theil inequality itself finds all its interest in the information-theoretic idea following that of main components (Kullback [22]) and based on the three axioms (Zero-valuation of certainty, Diminishing-valuation of probability, Additivity of independent events). A deep review of such of individual properties for a number inequality measures can be found in Cowell [13,23,24] for instance.
It is worth mentioning that the TLIM presented here is rather a mathematical form gathering of a number of different measures having different insights. Its main interest is to provide a general and uniform approach for dealing with both poverty and inequality measures in the same time and to avoid details and repetitions, in a coherent framework for useful comparison studies. In coming papers, the families presented by Cowell [13,23,24] will be studied in similar ways.
The motivations stated above lead to the study of the behavior of , will be an appropriate set of tools for the study of the influence of each measure on the other.
To achieve our goal we need a coherent asymptotic theory allowing the handling of longitudinal data as it is the case here and a stochastic process approach leading to asymptotic subresults with the help of the continuity mapping theorem.
We find that the functional empirical process, in the modern setting of weak convergence theory, provides that coherent asymptotic theory.
Indeed, we use bidimensional functional empirical processes and its stochastic Gaussian limit to entirely describe the asymptotic behaviour of in the Gaussian field of  and then find the law of  n s t as our best achievement.
The remainder of the paper is organized as follows. In Section 2, we remind key definitions and properties for functional empirical processes, and we state the asymptotic representation of the GPI of Sall and Lo [25] stated in Theorem 1 that will be used later on. In Section 3, we give our main results and make some commentaries and data driven applications to Senegalese pseudo-panel data are considered while the proofs and the tables are postponed in an appendix Section 5. Section 4 concludes.

Functional Empirical Process and
Representation of GPI

A Brief Reminder on Functional Empirical Processes
be a sequence of independent and identically distributed (i.i.d.) random elements defined on the probability space with values in some metric space . Given a collection of mesurable functions , the functional empirical process (FEP) is defined by: as consequences of the real Law of Large Numbers (LLN) and the real Central Limit Theorem (CLT).
When using the FEP, we may be interested in uniform LLN's and weak limits of the FEP considered as stochastic processes. This gives the so important results on Glivenko-Cantelli classes and Donsker ones. Let us define them here (for more details see [26,27]).
Since we may deal with non measurable sequences of random elements, we generally use the outer almost sure convergence defined as follows. Remind that a sequence converges outer almost surely to zero, denoted by whenever there is a measurable sequence of measurable random variables such that

Definition 2 A class is a Donsker class for
, or -Donsker class if converges in where is the Gaussian process, defined in Definition 2.


We will make use of the linearity property of both n and . Let be measurable functions satisfying (5) and , ,k The materials defined here, when used in a smart way, lead to a simple handling the problem tackled here.

Representation of the GPI
In this paper, we use the GPI in a unified approach that leads to an asymptotic representation for a large class of indices classified in three kinds.
First we consider the threshold condition: (H1) There exist 0 Based on these hypotheses, we put the reduced process of Sall et Lo (see [25]).
The representation results of [25] for the GPI is the following.
Theorem 1 Suppose that (H1)-(H6) are true, then we have the following representation Although these conditions may appear complicated, they are simple to check in usual cases with the popular poverty measures. We will see this in Section 3.
We are going to state our main results.

Notations
Let us consider the following Renyi representations. Let . Then we have the representation, meant as equalities in distribution: pose that is continuous. The copula associated with the couple Next we consider the bidimensional functional em- and the limiting centered Gaussian stochastic process its variance-covariance function defined by, for : , , where is the indicator function on the set

Main Theorems
We are now able to state our theorems. The first concerns the variation of the inequality measure.
The second concerns the variation of the GPI.
First of all, the results cover a large class of poverty measures and inequality indices. This explains why the notations seem heavy. Secondly, the variances of the limiting Gaussian processes seem also somehow tricky. But all of them are easily handled by modern computation means. We are going to particularise our results for famous measures and provide workable software codes for the computations.

Representation of Some Poverty Indices
We may easily find the functions g and  for the most common members of the GPI family (see [25,28]) in Table 1.

In this case, let
Where  When constructing pseudo-panel data, we get small sizes like n = 116. We use these sizes to compute the asymptotic variances in our results by mean of nonparametric methods. In real contexts, we should use high sizes comparable to those of the real databases, that is around ten thousands, like in the Senegalese case. Nevertheless, we back on medium sizes, for instance n = 696, which give very accurate confidence intervals.We use here the abreviations are given in Table 2.
The obtained confidence intervals are described in

Analysis
First of all, we find in Tables 3 and 4 in the appendix 5 that at an asymptotical level, all our inequality measures and poverty indices used here have decreased. When inspecting the asymptotic variance, we see in Table 4 that for the poverty indice, the FGT and the Kakwani classes respectively for 1   , 2   and k = 1, k = 2 have the minimum variance, specially for 2   and k = 2. This advocates for the use of the Kakwani and the FGT measures for poverty reduction evaluation. As for the inequality approach in Table 3, it seems that Atkinson measure ATK (0.5) has the minimum variance and then is recommended.
As for the ratio of the poverty index over the inequality measure, we have a dependence of over 50% for the following couples in Table 11, that we can find in Tables 5 to 8. The maximum ratio 3.024 is attained for FGT (0) and Atkinson (0.5). Based on these data, and on the confidence intervals in Table 9, we would report at least of 46.43% for these two measures and conclude that the gain over poverty in Senegal between these two periods is significally pro-poor. We would have worked with all couples with a ratio over 50% to have the same conclusion.
The present analysis should be developped in a separated paper research since this one was devoted to a theoritical basis. We plan to apply at a regional basis, that is for the countries of the UEMOA in West Africa.

Conclusion
We have been able to compute confidence intervals for the ratio of variations for the poverty and the inequality indices. The results enabled us to cheek whether the growth is pro or against poor in Senegal from 2002 to 2006. It always remains to undertake large scale data driven applications at a regional level, precisely in the UEMOA African area. We used in this paper a Theil-like family of inequality measures that does not include the celebrated and important Gini index. Moreover other the Theil-like families exist. It would be interesting to have the same theory developed here using the Gini index and other families as well. We plan to do it in a very close future.

Acknowledgements
We express our thanks to the Ministère de l'Enseignement Supérieur et de la Recherche for financial support under a FIRST grant 2013-2014.