1. Introduction
In this paper, we deal with a modern weak theory for some large class of inequality indices that, further, will allow to handle easy comparison studies with different kinds of statistics.
According to earlier economists, inequality indices are functional relations between the income and the economic welfare (see Dalton (1920)). This explains, among others, the wide variety of such indices in the literature (See, e.g., Cowell (1980a, b, 2000)).
Such statistics, of course, have been widely studied with respect to a great variety of interests, including statistical characterizations and asymptotic properties (See Davidson and Duclos (2000), Barrett and Donald (2009), for recent studies).
Recently, Greselin et al. (2009) provided a mathematical investigation of these indices in a modern setting including Vervaat processes, Lstatistics and empirical processes.
Having in mind the necessity of comparing inequality measures with different kind of statistics such as growth statistics, we aim at providing a coherent asymptotic weak theory for some class of inequality measures. Indeed we propose the functional empirical process setting (see Van der Vaart and Wellner (1996)) which provide natural Gaussian field in which many statistics used in Economics may be represented in.
Our best achievement consists of the asymptotic representations for the elements of our class of inequality measures, in terms of the above mentioned Gaussian field. The results are illustrated in data driven applications, on Senegalese data for instance.
The class on which we focus here is a functional family of inequality measures which gathers various ones around the central Theil measure. This class named after the TheilLike Inequality Measure (TLIM) will be the central point of our study. It includes the Generalized Entropy Measure, the Mean Logarithmic Deviation (Cowell (2003); Theil (1967); Cowell (1980a)), the different inequality measures of Atkinson (1970), Champernowne and Cowell (1998), Kolm (1976a), and the divergence of Renyi (1961).
This means that, here, we will not discuss other inequality statistics such as the Gini, the Generalized Gini, the SGini, the EGini (See Barrett and Donald (2009)). Those statistics and similar ones will be treated in separate papers.
Now we are going to introduce our family. For that, let
denote the income (or expense) random variable related to a given population. We assume that
and its independent observations are defined on the probability space
and take their values in on some intervaland have common cumulative distribution function
(cdf), , . In this paper, we only use LebesgueStieljes integrals and for any measurable function , we have, whenever it makes sense,where is the measure image of by , but is also LebesgueStieljes probability measure characterized by: for any .
Now, consider a sample of individuals or households of that population and observe their income . We define the following family of inequality indices, indexed by as follows
(1.1) 
where is the empirical mean while , , and are real and measurable functions of and . The exact form of is not important here, in opposite to the conditions on the functions , , and under which the results are valid. In a future paper on the uniform limits in , that class will be crucial.
We will see below that under specific hypotheses on and , converges to the exact inequality measure
(1.2) 
where is the mathematical expectation of that we suppose finite here. We will come back later on the function classes and in which and are supposed to lie.
Each measure of this Theillike family has its own particular properties, that are derived from the combination of different concepts. One may mention the concept of welfare criteria (Atkinson (1970), Sen (1973)), that of the analogy with analysis of risks (Harsanyi (1953), Harsanyi (1955), Rothschild and Stiglitz (1973)), that of the complaints approach (Temkin (1993)) etc.
The Theil inequality itself finds all its interest in the informationtheoretic idea following that of main components (Kullback 1959). It is based on the three following axioms: Zerovaluation of certainty, Diminishingvaluation of probability, Additivity of independent events. A deep review of such of individual properties for a number inequality measures can be found in Cowell (Cowell (1980a, b, 2000)) for instance.
It is worth mentioning that the TLIM presented here, is rather a mathematical form gathering a number of different measures.
The rest of our paper is organized as follows. In Section 2, we describe the TLIM family and show how the particular indices are derived from it. In Section 3, we briefly recall the functional empirical processes setting. In section 4, we deal with the asymptotic theory of the TLIM, state and describe our main results and demonstrate them. Section 5 is devoted to datadriven applications. We finish by a conclusion in Section 7.
2. Description of the TLIM
This inequality measures mentioned above are derived from (1.1) with the particular values of the mesurable functions and as described below for all
2.1. Generalized Entropy
2.2. Theil’s measure
2.3. Mean Logarithmic Deviation
2.4. Atkinson’s measure
2.5. Champernowne’s measure
2.6. Kolm’s measure
2.7. Divergence of Renyi
3. The functional empirical process
Let be a sequence of independent and identically distributed random elements defined on the probability space with values in some metric space Given a collection of measurable functions satisfying
where is the mathematical expectation of , the functional empirical process (FEP) based on the and indexed by is defined by:
This process is widely studied in Van der Vaart and Wellner (1996)
for instance. It is readily derived from the real Law of Larges Numbers (
LLN) and the real Central Limit Theorem (
CLT) that and that , where(3.1) 
whenever .
When using the FEP, we are often interested in uniform LLN’s and weak limits of the FEP considered as stochastic processes. This gives the so important results on GlivenkoCantelli classes and Donsker ones. Let us define them here (for more details see Van der Vaart and Wellner (1996)).
Since we may deal with non measurable sequences of random elements, we generally use the outer almost sure convergence defined as follows:
a sequence converges outer almost surely to zero, denoted by whenever there is a measurable sequence of measurable random variables such that
The weak convergence generally holds in the space of all bounded real functions defined on equipped with the supremum norm
Definition 1.
is called a GlivenkoCantelli class for , if
Definition 2.
is called a Donsker class for , or Donsker class if converges in to a centered Gaussian process with covariance function
Remark 1.
When and , is called real empirical process and is denoted by
In this paper, we only use finitedimensional forms of the FEP, that is And then, any family of measurable functions satisfying (3.1), is a GlivenkoCantelli and a Donsker class, and hence
where is the Gaussian process, defined in Definition 2. We will make use of the linearity property of both and . Let be measurable functions satisfying (3.1) and , then
The materials defined here, when used in a smart way, lead to a simple handling of the problem which is addressed here.
4. Our results
Let us introduce some notation.
for all , we define the following function
with and is the derivative of the function
The following general condition will be assumed in all the paper:

is not null in a neighborhood of
Here are our main results.
4.1. Pointwise asymptotic laws
Consider the following hypotheses based on the functions . The A1.x series concern the almostsure limits and the A2.x the asymptotic normality.


is a continuous function on

for and is continuous on


is continuously differentiable such that

is continuously differentiable at
We have :
Theorem 1.
Suppose that the conditions and are satisfied, then converges almost surely to
Theorem 2.
Suppose that the conditions and are satisfied, and is finite. Then
(a) we have the following asymptotic representation in the empirical functional process
where
(b) and we have the convergence in distribution, as tends to infinity, of to centered normal Gaussian law:
where
with
Remark. The main result is the one given in Point (a). From it, Point (b) is deduced in a straightforward way.
The results above cover all the TLIM class. They should be particularized for the practitioner who would pick one of the elements of that class for analyzing data. Here are then the details for each case.
4.2. Particular cases for pointwise results
a. The Theil’s measure
The empirical form of Theil measure is defined as follows
Denote by
the continuous form of the Theil measure.
All these functions are continuous on then the assumptions defined above become for the a.s. requires that is finite and . As for the asymptotic normality, we need that
And we have . We conclude that
with
b. The Mean Logarithmic Deviation
Let
be the empirical form of the Mean Logarithmic Deviation. Its theoretical form is given as folllows
These specific functions are given by:
The consistency requires that and that while the normality is got when
In that case, we find easily that and
where
c. The Champernowne’s measure
In this case, the specific functions are given by:
And, the various forms are:
We find that and where is the Mean Logarithmic Deviation.
As is continuous on we consider the same hypotheses as in the case of Mean Logarithmic Deviation.
The function is continuously differentiable, we put and then we have
with
d. Cases of the Generalized Entropy the Atkinson’s measure the Divergence of Renyi
We may gather these indices into one subclass by giving different values to the function and to the parameter , with this common expression
and then give a general description of the results. For that, Let and
We require for consistency that and that and, for asymptotic normality that
Further, let and . Then we get
which tends towards a centered Gaussian process with variance
(4.1) 
Now, we may return to the individual cases.
d.1. Generalized Entropy
We find , from there, we get the variance
d.2. Atkinson’s measure
Put . We similarly get that
d.3. Divergence of Renyi
By taking we obtain by the same way, that
where is given in (4.1).
e. Case of the Kolm’s measure
This index is defined for and its specific functions are:
Its empirical form is given by
and its theoretical form is defined as follows
We need for consistency that and that and, for asymptotic normality that
and are finite.
Then we have and
Put
Then
Since we deduce that
Finally, we summarize the used abbreviations in Table 1, and, for each index, the expression of the function and in Table 2 where we can find the expressions of and
Notations  Indices 

Generalized Entropy with parameter  
THEIL  Theil 
MLD  Mean Logarithmic Deviation 
ATK and  Atkinson with parameter 
CHAMP  Champernowne 
KOLM  Kolm with parameter 
DR  Divergence of Renyi with parameter 
Indices  

THEIL  
MLD  
ATK  
CHAMP  
KOLM  
DR 
5. Proof of Theorems 1 and 2
Proof of Theorem 1.
On one hand, denote by
(5.1) 
by decomposing the difference of and we get the next equality
As for all the function is continuous on and using the fact that converges almost surely to then we have when tends to infinity
(5.2) 
We have also
Or the sequence of the random variables is independent and identically distributed, and as by the hypothesis then the Law of Large Numbers implies that
(5.3) 
On the other hand, as satisfies the hypothesis then we deduce that
Proof of Theorem 2.
Using the equation (5.1), we have
Since is continuously differentiable at for we get
Then
But
then we get the next expression
Then
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