A Glivenko-Cantelli Bootstrap Theorem for the Foster-Greer-Thorbecke Poverty Index

We assume the Foster-Greer-Thorbecke (FGT) poverty index as an empirical process indexed by a particular Glivenko-Cantelli class or collection of functions and define this poverty index as a functional empirical process of the bootstrap type, to show that the outer almost sure convergence of the FGT empirical process is a necessary and sufficient condition for the outer almost sure convergence of the FGT bootstrap empirical process; that is: both processes are asymptotically equivalent respect to this type of convergence.


Introduction
The problem of estimating one-dimensional poverty measures is theoretically addressed in this paper, developing a characterization in law of large numbers, in the framework of bootstrap empirical processes. To achieve this goal, first we introduce some basics elements: let N be a statistical universe of individuals (let us say households), such that for each one of them it is possible to determine its level of income, following e.g. [12,17], for a random sample of n individuals withdrawn from this population, a measure or classic index of poverty is a function P : R n+1 + → [0, 1], where the value of P(y, z) indicates the degree or level of poverty associated with the vector of incomes y = (y 1 , y 2 , . . . , y n ) ∈ R n + and the fixed poverty line z ∈ R + , such that any j-th individual of the random sample is considered poor if y j < z.
This type of measures is commonly denominated one-dimensional poverty indices because in their construction only one economic dimension is considered. With the research published by Sen in 1976 about the first axioms or properties of the axiomatic method of poverty (see [18]), the idea of studying this problem as a phenomenon that depends only on the income acquires greater mathematical rigor within the economic theory, and various measures of poverty begin to be proposed, all of which are supported in the Sen's axiomatic definition. In this approach, one of the most important measures is the Foster-Greer-Thorbecke (FGT) poverty index (1984, [7]): that emphasizes the degree of aversion to poverty by including the parameter α ≥ 0, where according with [12], [17] and [22] among others, the sum in (1) is only over q: the number of poor individuals for the random sample 1 .
On the other hand, the bootstrap technique was introduced by Efron in 1979 and 1982 [5,6], as a method to estimate the sample distribution of a statistics. In general, let Y 1 , Y 2 , . . . , Y n be a finite collection of i.i.d. random variables with law of probability P, if θ := θ(P) is a parameter of interest, θ n := θ n (Y 1 , Y 2 , . . . Y n ; P) an estimator of θ, andŶ 1 ,Ŷ 2 , . . . ,Ŷ n an i.i.d. random sample with replacement of the empirical probability measure P n . Then, the "bootstrap principle" consists in estimating the unknown distribution of θ n throughθ n := θ n (Ŷ 1 ,Ŷ 2 , . . . ,Ŷ n ; P n ).
In 2009, Lo and Seck found that the FGT poverty index defined in (1) understood as an empirical process satisfies a very particular law of large numbers (see [14]). Now, we found that it is possible to establish an important 1 For example, if α = 0 the index is interpreted as the incidence of poverty, while for α = 1 and α = 2 is interpreted as the intensity or severity of poverty and the depth or inequality among the poor, respectively. For a detailed discussion about the axiomatic method and all the one-dimensional poverty indices proposed in the literature, see e.g. [22].
Volumen 54, Número 2, Año 2020 G a l l e y p r o o f convergence relationship between the FGT empirical process of Lo and Seck and another one of the bootstrap type defined below. The statements of our main result presented in the paper are inspired (among others) in the theorems 3.3 of [21] and 10.15 of [13], that succinctly tell the reader: if one wants to obtain a uniform bootstrap approximation one should check if a certain class is Glivenko-Cantelli. Indeed, the theoretical proposal presented here is a particular contribution over the literature: formally states that under certain conditions, the FGT empirical process considered as an estimator of the average poverty level (statistics) converges almost surely to the real and unknown average poverty level (parameter) reflected in the mean function of the corresponding process, if and only if the FGT bootstrap empirical process considered as a bootstrap estimator of the average poverty level (bootstrap statistics) converges almost surely to the correspondent estimator, for a random sample of incomes statistically large and representative of a statistical universe of households.
The article is structured in four Sections, including this introduction. In Section 2, we present the problem statement. Consequently, Section 3 presents the main result, and finally, the Section 4 contains all the tools required for its development.

The problem
Consider the product probability space (Ω, Σ, P ) := (X N , A N , P N ). In this framework X N is the sample space of all infinite-numerable sequences of incomes, such that for any infinite-numerable sequence ω := (y 1 , y 2 , . . .) ∈ X N , we can define a function or coordinate projection Y : X N → X such that Y (ω) = y ∈ X , with probability distribution function F(z) = P(Y ≤ z) for z ∈ R + fixed. Moreover, according with [3] and [9] among others, we can define a finite collection of functions Y 1 , Y 2 , . . . , Y n i.i.d.∼ P, so that for each j ∈ N, Y j : X N → X is the j-th coordinate projection on (X N , A N , P N ), such that for all ω := (y 1 , y 2 , . . .) ∈ X N , Y j (ω) = y j ∈ X , with empirical distribution function: for z ∈ R + fixed, where q = nF n (z). Particularly, the i.i.d. collection {Y j } n j=1 is an empirical process of n random variables, where each projection Y j on the product probability space (X N , A N , P N ) represents the observed level of income for the j-th statistical individual of the random sample of size n in the probability space (X , A, P). Let P n : X N × A → [0, 1] be the empirical measure associated with this sequence of random variables, where:  [14]) define the class or collection of functions F Γ := {f α , α ≥ 0}: which allows to describe the FGT poverty index defined in (1) like the func- for all α ≥ 0, j = 1, 2, . . . , n. LetP be the classical Efron nonparametric bootstrap empirical measure, where it is possible to consider n bootstrap samplesŶ 1 ,Ŷ 2 , . . . ,Ŷ n of a determined collection of i.i.d.∼ P functions or coordinate projections Y 1 , Y 2 , . . . , Y n . Following e.g. [2,15,21], we can consider a triangular array of exchangeable random variables W := {W nj : j = 1, 2, . . . , n, n = 1, 2, . . .} defined on (W, D, P W ), such that these random variables can be interpreted as random weights, in the sense that each component W nj reflects the number of times that Y j is selected for the n trials of the bootstrap sample with replacement, where: is just the exchangeably weighted bootstrap empirical measure, such that the classical measureP n defined above is a special case ofP W n obtained by taking (W n1 , W n2 , . . . , W nn ) = W n = M n , with M n = (M n1 , M n2 , . . . , M nn ) ∼ M ult n (n, (1/n, 1/n, . . . , 1/n)). Consequently, we can define the functional or We consider a exchangeably weighted version of the bootstrap in this paper, because under the hypothesis of "exchangeability", we can "emulate" the Strobl's lemma 4.9 of the In [14], Lo and Seck show that the class F Γ is strong P-Glivenko-Cantelli; that is: as n → ∞, where: is the correspondent mean function of the FGT empirical process P n (f α ) defined above. Now, we found that under the G.-C. hypothesis, the trajectories or realizations of P W n get uniformly closer to P n as n → ∞, and that the reciprocal is also true.
A2. W nj ≥ 0 for all n, j, and n j=1 W nj = n, for all n.
Using the last condition like in [21], page 598: That is, a multiplier process with ξ nj := W nj −1 and Z j := δ Yj −P, respectively, . Particularly, we redefined A1 and considered a couple of additional conditions for the weights ξ nj : Section 4 for the correspondent bootstrap processes in our main result, respect to the Σnmeasurability and the backward submartingale property required in the Strobl's result, with Σn being the filtration defined in remark 4.7 of this Section. Additionally, following e.g. [21], our main result developed in Section 3 will allow to present in future papers, at least one Glivenko-Cantelli theorem for the classical nonparametric bootstrap empirical measurê Pn := 1 n n j=1 M nj δ Y j as a direct consequence of this main result (see e.g. the theorem 3.3 for the exchangeable bootstrap and Theorem 3.2 for Efron's bootstrap, that follows as a corollary of the first mentioned here in [21]).

Revista Colombiana de Matemáticas
G a l l e y B3. ξ n1 satisfies the weak second-moment condition: for all y j ∈ X and the fixed poverty line z ∈ R + . Then, the following are equivalent: Proof. (i)⇒(ii): We know thatP W n − P n = 1 n n j=1 ξ nj Z j , by (10). It follows: for any 1 ≤ n 0 < n. This is the right side in the lemma of inequalities for the bootstrap process (lemma 4.12). We point out a few properties related to this upper bound: (a) Since F Γ is strong P-Glivenko-Cantelli by hypothesis; i.e., the measurable cover function of F Γ defined above, by the inequality (19) of the Strobl's theorem for backward submartingales (lemma 4.9), it follows that Emulating the Strobl's theorem for { P n * FΓ } n∈N , we can conclude that this process is a backward submartingale respect to {Σ n } n∈N , the σ-algebra defined in Remark 4.7. Since P n * FΓ · I{ P n * that is, E * P W n − P n FΓ → 0, as n → ∞. This implies that P W n − P n * FΓ P − → 0, because the convergence in outer mean implies convergence in outer probability.
where the right side of (12) is integrable by remark 4.5. Using this fact and the condition of exchangeability B1 for the random weights ξ nj , we can emulate again the Strobl's theorem to see that { P W n − P n * FΓ , Σ n } n∈N is a backward submartingale. By the inequality (12), is clear that sup and from the Doob's theorem applied above, it follows that P W n − P n * FΓ converges almost surely to a finite limit, but P W n − P n * FΓ P − → 0, thus this limit must be equal to zero and then P W n − P n FΓ From the upper bound of (24) in the lemma 4.10 of Rademacher symmetrization, it follows: that is, E * P n − P FΓ → 0, when n → ∞. This implies P n − P * FΓ P − → 0 using the convergence argument discussed above, we can conclude that F Γ is weak P-Glivenko-Cantelli.
To finish the proof, the process { P n − P * FΓ , Σ n } n∈N is a backward submartingale by the Strobl' theorem 4.9, and E( P n − P * FΓ ) ≤ 2P(F * ) < ∞, by inequality (20), for each n ∈ N. Consequently, sup n∈N E( P n − P * FΓ ) < ∞, and from the Doob's theorem, it follows that P n − P * FΓ converges almost surely to a finite limit, but P n − P * FΓ P − → 0, thus this limit must be equal to zero, and then P n − P FΓ a.s. * − −− → 0. Note 1. Following page 15 in [8], if we assume measurability for the collection {Y j } n j=1 , then the index " * " can be removed in the notation, and all the results presented here also hold (see e.g. the definition 4.4 in the next Section). Lemma 4.6 is similar to Lemma 8.13, page 141 in [13], applied now to the classes of functions F Γ andḞ Γ , respectively. Lemma 4.9 is basically the original Theorem 1.1, pages 826-829 in [19], for the class F Γ . Lemma 4.10 corresponds to Lemma 11.2.12, pages 343-344 in [4], applied to the class F Γ . The lemma 4.12 developed for this collection, is similar to the Lemma 2.9.1, pages 177-179 in [20]; or Lemma 2.2, pages 595-596 in [21]. Lemma 4.13 is similar to Lemma 4.7, page 2071 in [15], considering the random weights ξ nj . For details about all the proofs see [11], and for a detailed discussion about the bootstrap see e.g. [10]. (1) {Y n } n∈N is adapted to the filtration {Σ n } n∈N (or more generally, adapted to the P -completion of {Σ n } n∈N ). That is, Y n is Σ n -measurable for each n ∈ N (or more generally, measurable for the P -completion of {Σ n } n∈N );

Tools Required for the Main Result
(2) E(|Y n |) < ∞, for each n ∈ N; Then {Y n , Σ n } n∈N is said to be a reversed or backward submartingale. integrable random variable Y such that: Proof Then this process is convergent in mean; that is: where f * : Ω → R is the minimal measurable majorant or smallest measurable function above f , that satisfies: (1) f * (ω) ≥ f (ω) for each ω ∈ Ω; (2) For any measurable function g : Ω → R with g ≥ f a.s., f * ≤ g a.s. Remark 4.5. Consider the class F Γ := {f α , α ≥ 0}, and let F : X → R be the envelope function of this collection, with F := f α FΓ such that |f α (y)| ≤ F (y) = sup fα∈FΓ |f α (y)| for each y ∈ X and f α ∈ F Γ . Now we can define the clasṡ According to the above definition, under integrability, the outer expectation of an envelope function for a determined class or collection is equal to the expected value of the measurable cover function respect to this envelope. In other terms, if P * (F ) < ∞ and P * (Ḟ ) < ∞, then is true that P * (F ) = P(F * ) and P * (Ḟ ) = P(Ḟ * ), where F * := ( f α FΓ ) * anḋ F * := ( f α − P(f α ) FΓ ) * are the measurable cover functions for F andḞ of the classes F Γ andḞ Γ , respectively. In this setting, F ≤ F * , F * is measurable, and F * ≤ h P -a.s. for all measurable function h ≥ F . The same analysis follows foṙ F andḞ * .
Remark 4.7. Let P n be the n-th empirical measure on (Ω, Σ, P ) := (X N , A N , P N ), it follows that the class C of all sets invariant under permutations of the n first coordinates Y j : X N → X is a σ-algebra 3 . Specifically, we can define Σ n as the smallest σ-algebra that contains all the sets {A ∈ Σ : I A (y) = I A (πy)}; that is, invariants under any permutation π ∈ S(n) of the first n coordinates Y j : X N → X , such that Σ n ⊃ Σ n+1 , for each n ∈ N.
Applying the same argument as above, is clear that if y ∈ Y j (A), then y ∈ Y j (A). Therefore Y j (A) = Y j (A), and it follows that: If F Γ has a measurable cover function F * ∈ L 1 (X , A, P), then { P n − P * FΓ , Σ n } n∈N is a backward submartingale; that is, P n − P * FΓ is Σ n -measurable, P -integrable, and for each n ∈ N.
Since P n − P FΓ is invariant under all permutations of the first n coordinates, for all π ∈ S(n), and then min π∈S(n) where the left side of the inequality is a Σ-measurable function. Therefore: min π∈S(n) g • f π ≥ P n − P * FΓ = g P -a.s., by the definition of measurable cover functions. Thus: min π∈S(n) g • f π = g P -a.s.
then P n,n+1 ≡ P n := 1 n n j=1 δ Yj , and for j = n + 1, P n,j has the same properties as P n . Therefore: The right side of (22) is Σ-measurable, so it is an upper bound of the outer empirical discrepancy P n+1 − P * FΓ P -a.s., too. Therefore, Thus, it is enough to prove that for each 1 ≤ j ≤ n + 1, For m ∈ N fix but arbitrary, let h : X → R such that However for all A ∈ Σ n+1 . Thus, we can use the argument developed in Remark 4.8 to verify (17), and obtain: using the fact that the sets A in Σ n+1 are invariants under all permutations of the first n + 1 coordinates. Hence, the equality (23) is true, and consequently the backward submartingale property (18) is satisfied.
Lemma 4.12 (Inequalities for the bootstrap process with exchangeable multiplier random weights). Given (X N , A N , P N ) × (W, D, P W ) × (Z, C, P ), the basic product probability space. Let {Z j } n j=1 be an i.i.d. empirical process such that E * Z j FΓ < ∞ for each j ≤ n, independent of the i.i.d. Rademacher collection { j } n j=1 . Then, for a collection of i.i.d. exchangeable random weights {ξ nj } n j=1 with ξ n1 2,1 < ∞ and E(ξ nj ) = µ, independent of the collection {Z j } n j=1 and any 1 ≤ n 0 < n, For symmetrically distributed variables ξ nj around µ, the constants 1/2, 2 and 4 can all be replaced by 1, and µ in the left-hand side of (26) is equal to zero.