Statistical estimation of gap of decomposability of the general poverty index

For the decomposability property is very a practical one in Welfare analysis, most researchers and users favor decomposable poverty indices such as the Foster-Greer-Thorbeck poverty index. This may lead to neglect the so important weighted indices like the Kakwani and Shorrocks ones which have interesting other properties in Welfare analysis. To face up to this problem, we give in this paper, statistical estimations of the gap of decomposability of a large class of such indices using the General Poverty Indice (GPI) and of a new asymptotic representation Theorem for it, in terms of functional empirical processes theory. The results then enable independent handling of targeted groups and next global reporting with significant confidence intervals. Data-driven examples are given with real data.


Introduction
We are concerned in this paper with the statistical estimation of the gap of decomposability of the class of the statistical poverty indices in general. Suppose that we have some statistic of the functional form J n = J(Y 1 , ..., Y n ) where E = {Y 1 , ..., Y n } is a sample of the random variable Y defined on a probability space (Ω, A, P) and drawn from some specific population. Now, suppose that this population is divided into K subgroups S 1 , ..., S K and let us, for each i ∈ {1, ..., K}, denote the subset of the random sample {Y 1 , ..., Y n } coming from S i by E i = {Y 1,i , ..., Y n i ,i } and then put J n i (i) = J(Y 1,i , ..., Y n i ,i ). The statistic J n is said to be decomposable whenever one always has whatever may be the way in which E is partitioned into the E i 's (i = 1, ..., K). This property is a very practical one when dealing with the poverty measures or welfare measures in general for the following reason. If we are willing to monitor the poverty situation, it may be very useful to target some sensitive areas or subgroups. By dividing the population into targeted groups, and estimating the poverty intensity by J n i (i) (resp. variation of poverty by ∆J n i (i)) in each group, one would be able to report the poverty intensity (resp. global poverty variation) by (1.1) (resp. ∆J n = 1 n K i=1 n i ∆J n i (i)), provided that the samples are the same as it is the case in longitudinal data. Thus, decomposability allows an independent handling of poverty for different areas and next an easy reconstruction of the global situation. Now in the specific case of poverty indices, we mainly have the non-weighted ones and the weighted ones. The statistics in the first case are automatically decomposable and then are mostly preferred by users. However, the weighted measures, which in general are not decomposable, have very interesting properties in poverty analysis. Dismissing them only for nondecomposability would result in a disaster. We tackle this problem in this paper. Indeed, by estimating the following gap of decomposability with significant confidence intervals, we would be able to handle separated analyses in the subgroups and report the global case and, at the same time, make benefit of the other properties of such statistics.
The remainder of the paper is organized as follows. In Section 2, we give a brief introduction of the poverty measures and to the General Poverty Index (GPI). In Section 3, we return back to the decomposability problem by describing the drawing scheme under which the results are given. In Section 4, we state the results which are applied to the Senegalese and Mauritanian data in Section 5. The proofs are given in Section 6. The concluding remarks are in Section 7. The paper is finished by an appendix in Section 8.

A brief reminder on Poverty measures
We consider a population of individuals or households, each of which having a random income or expenditure Y with distribution function G(y) = P(Y ≤ y). In the sequel, we use Y as an income variable although it might be any positive random variable. An individual is classified as poor whenever his income or expenditure Y fulfills Y < Z, where Z is a specified threshold level (the poverty line).
Consider also a random sample Y 1 , Y 2 , ...Y n of size n of incomes, with empirical distribution function G n (y) = n −1 # {Y i ≤ y : 1 ≤ i ≤ n}. The number of poor individuals within the sample is then equal to Q n = nG n (Z). And, from now on, all the random elements used in the paper are defined on the same probability space (Ω, A, P).
Given these preliminaries, we introduce measurable functions A(p, q, z), w(t), and d(t) of p, q ∈ N, and z, t ∈ R. Set B(Q n ) = Qn i=1 w(i).
Let Y 1,n ≤ Y 2,n ≤ ... ≤ Y n,n be the order statistics of the sample Y 1 , Y 2 , ...Y n of Y . We consider general poverty indices (GPI) of the form (2.1) where µ 1 , µ 2 , µ 3 , µ 4 are constants. This global form of poverty indices was introduced in [15] (see also [13], [15] and [16]) as an attempt to unify the large number of poverty indices that have been introduced in the literature since the pioneering work of the Nobel Prize winner, Amartya Sen(1976) who first derived poverty measures (see [19]) from an axiomatic point of view. A survey of these indices is to be found in Zheng [24], who also discussed their introduction, from an axiomatic point of view. We will cite a few number of them here just to make clear the minds and prepare the data-driven applications in Section 5.
One may devide the poverty indices into two classes. The first includes the nonweighted ones. The most popular of them is the Foster-Greer-Thorbecke(1984) [7] class which is defined for α ≥ 0, by For α = 0, (2.2) reduces to Q n /n, the headcount of poor individuals. For α = 1 and α = 2, it is respectively interpreted as the severity of poverty and the depth in poverty. (2.2) is obtained from (2.1) by taking δ = I d , w ≡ 1, d(u) = u α , B(Q n , n) = Q n and A(Q n , n, Z) = Q n .
Next, we have for α ≥ 0, the Chakravarty family class of poverty measures is obtained from (2.1) by taking Y α and Z α as respectively transformed income Y and threshold Z and The statistics in this class are decomposable and are not concerned by the present work.
The second class consists of the weighted indices. We mention here two of its famous members. The Sen(1976) index (see [19]) B(Q n ) = Q n (Q n + 1)/2, µ 1 = 0 and µ 3 = µ 2 = µ 4 = 1. The Shorrocks(1995) index (see [21]) is obtained from (2.1) by taking B(Q n , n) = Q n (Q n + 1)/2, A(n, Q n , Z) = Q n (Q n + 1)/2n, 3) and (2.4) evaluate the poverty intensity by giving a more important weight on the poorest individuals. This means that a small decrease of the intensity on the poorest household indicates significant improvement in the population.
In the applications, we mainly deal with these two specific measures because of their importance in poverty analysis. Notice that the Thon measure ( [22]) is different from the Shorrocks one only by their normalization coefficients which are respectively n(n+1) and n 2 , so that they have the same asymptotic behavior. Finally, we have the following generalization of the Sen measure given by Kakwani(1980) [11], where k is a positive parameter. Notice that J n (1) is the Sen measure. Notice also that, under mild conditions, J n converges in probability to the Exact General Poverty Index (EGPI) (see [1], [2], [3] and [13]), where L 1 is some weight function depending on the distribution function. This result will be proved again in Theorem 1 below.

Statistical decomposability
From now, we suppose that our studied population of households is divided into K subgroup such that, for each i ∈ {1, ..., K}, the probability that a randomly drawn household comes from the i th subgroup is p i > 0, with p 1 + ... + p K = 1. Let us suppose that we draw a sample of size n from the population : Y 1 , ..., Y n and let us denote those of the n * i observations coming from the i th subgroup, (1 ≤ i ≤ K) by Y i,j , j = 1, ..., n * i . Let J n * i (G i ) = J n * i (Y i,1 , ..., Y i,n * i ) the empirical index measured on the i th subgroup and J n (G) the global index. Clearly, decomposability implies for all n ≥ 1, Surely, n * = (n * 1 , ...n * K ) follows a multinomial law with parameters n and p = (p 1 , ..., p K ). Since each p i > 0, we have that for each 1 ≤ i ≤ K, n * i → ∞ a.s., as n → ∞. We will have by (1.1) and by (2.5), The right member of this equation is the exact gap of decomposability gd. It follows that gd is zero if the distribution of the income is the same over all the population, that the more homogeneous the income is over the population, the lower the gap of decomposability gd is. As a first result, we get that the decomposability does not, asymptotically at least, matter for a more or less homogeneous population. That is, the decomposability is not only a functional form matter (of the index), but it is also a statistical one since whatever might be the index, decomposability is asymptotically obtained when the subgroups have the same distribution. For example, it has been pointed out in ( [10]), for the Senegalese poverty databases from 1996 to 2001, that the gaps of decomposability were very low for various stratifications (in regions, gender, ethnic groups, etc.). The apparent reason was the homogeneity of the income. Such results are confirmed in Section 5.
Now we want to find the law of for a more accurate estimation of gd by confidence intervals. At this step, we have to precise our random scheme. We put a probability space (Ω 1 × Ω 2 , A 1 ⊗ A 2 , P 1 ⊗ P 2 ) and put P = P 1 ⊗ P 2 . We draw the observations in the following way. In each trial, we draw a subgroup, the ith subgroup (E i ) having the occurring probability p i . And we put π i,j (ω 1 ) = I (the i th subgroup is drawn at the j th trial) (ω 1 ), 1 ≤ i ≤ K, 1 ≤ j ≤ n. Now, given that the i th subgroup is drawn at the j th trial, we pick one individual in this subgroup and observe its income Y j (ω 1 , ω 2 ). We then have the observations We have these simple facts. First, for 1 ≤ i ≤ K, Secondly, the distribution of Y j given (π i,j = 1), is G i , that is We conclude that {Y 1 , ..., Y n } is an independent sample drawn from G(y) , the mixture of the distribution functions of the subgroups incomes. Finally, we readily see that conditionally on n * ≡ (n * 1 , n * 2 , ..., n * K ) = (n 1 , n 2 , ..., n K ) ≡ n with n 1 + n 2 + ...

Our results
The results stated here hold for a very large class of poverty measures summarized in the GPI. This is why we need the representation Theorem of the GPI in [18]. In fact, we do not need here the complete form of [18], but a special case of it, based on the assumptions described below. For that, suppose that G i (1 ≤ i ≤ K), is the distribution function of the income for the ith subgroup, and G is the distribution function of the income for the global population. Let also γ(x) = d Z−x Z 1 (x≤Z) and e(x) = 1 (x≤Z) . The following assumptions are required.
(HD4) For a fixed x, the functions y → ∂c ∂y (x, y) and y → ∂π We also need the following definitions, for G 0 ∈ {G, G 1 , ..., G K }, with the conventions that for G 0 = G, we denote g 0 = g and ν 0 = ν. For We are now able to briefly describe the approximation of [18] : if G 0 fulfills (HD1), ..., (HD6), then as n → +∞, we have is the functional empirical process and is a residual stochastic process introduced in [18] and widely studied in [12], where G n is the empirical distribution function associated with {V 1 , ..., V n } sampled from G 0 .
We are now able to state our main result.
Remark 1. This clearly makes the so important decomposability requirement less crucial since the default of decomposability may be estimated by confidence intervals based on this theorem, as we showed it in the next section.

Kakwani case.
We also have the same conclusion for the Kakwami measure of parameter k ≥ 1 with c(x, y) = (x − y) k and π(x, y) = y k /x,

5.4.
Data-driven applications. In this note, let us focus on the Sen case, which is more tricky than the Shorrocks one. We consider the Senegalese database ESAM 1 of 1996 which includes 3278 households. We first consider the geographical decomposition into the areas, Dakar is the Capital. We have the Sen measure values for the whole Senegal and for its ten sub-areas. Let us compute the different variances ϑ 2 1 ϑ 2 2 and ϑ 2 3 of Theorem 1 with the empirical estimations p i ≈ n i /n,. We obtain for the geographical decomposability in Senegal : ϑ 2 1 + ϑ 2 2 = 0.093195, ϑ 2 1 + ϑ 2 3 = 0.093224 and gd n = 1.25450 10 −3 . This gives the 95%-confidence : We remark the very accurate estimation of the Sen index for the whole country of Senegal which makes us tell that this index is practically decomposable in this empirical case. We have already explained that decomposability does not matter when the distribution is uniform in the population. It happens that earlier works show that the senegalese date are well fitted by the lognormal or the Singh-Maddala model for each area with very similar parameters. Now for a decomposition with respect to the household chief gender, we get the sen measure values. Our general conclusion is that for all these cases, the sen measure is almost decomposable. But, this does not really matter. The important result is that we are able to have an accurate estimation of the gap of decomposability.

Proofs
To begin, we need more notations to describe the representation result of [18], in an appropriate way to our proof. Let G 0 ∈ {G, G 1 , ..., G K } and let a sample of incomes {V 1 , ..., V m } from G 0 . Let α G 0 ,m the uniform empirical functional process based on and define an other empirical process, called here residual empirical process, where G G 0 ,m is the empirical distribution function associated with {V 1 , ..., V m }. The representation Theorem of Sall and Lo [18] establishes under the hypotheses (HD0)-(HD6), for J( as m → ∞, where g 0 and ν 0 are described in (4.1) and (4.5).
Before going any further, we should precise the notations for the global population and the subgroups. For G = G 0 , we drop the subscript G 0 so that α n , β n , G n , J n are respectively the empirical, the residual empirical process (6.1), the empirical distribution function and the GPI based on the sample Y 1 , ..., Y n , and J = J(G) = H c (G)/H π (G). As well the functions g 0 and ν 0 are denoted as g and ν for G = G 0 . For G = G i , 1 ≤ i ≤ K, we use the subscript i so that α i,n * i , β i,n * i , G i,n * i , J i,n * i will respectively denote the empirical, the residual empirical process (6.1), the empirical distribution function and the GPI based on the sample Y i,1 , ..., Y i,n * i , and J i (G i ) = H c (G i )/H π (G i ), accordingly to the notations of Section 4, and the functions g 0 and ν 0 are denoted as g i and ν i in this case. But sometimes we may feel the notations so heavy and then lessen them. For example, we only put To begin the proof, we remark that n * (ω 1 ) = (n * 1 (ω 1 ), ..., n * K (ω 1 )) → P 1 {+∞} K as n = n * 1 (ω 1 ) + ... + n * K (ω 1 ) → ∞. We then get (6.2) √ n(J n (G) − J(G)) = α n (g) + β n (ν) + o P (1) := γ n + o P (1) and for any 1 ≤ i ≤ K, Now we use the intermediate centering coefficient to get from (6.2) and (6.3) → P 1 ⊗P 2 0, as n → ∞. Then, we have Remark that
Recall that, by the classical limiting law of the multinomial K-vector, We remark that this is the variance of the function exp(htD * (n)/(n * = n))P (n * = n) By putting together the previous formulas, and by letting ε ↓ 0, we arrive at ψ d * * n (t) → exp(−(ϑ 2 1 + ϑ 2 2 )t 2 /2). This proves the asymptotic normality of dg * n of the theorem corresponding to S * * n . That of dg * n,0 corresponds to S * n . This latter is achieved by omitting the term (6.13). This leads to M h obtained from F h by dropping J i (G i ). This completes the proofs.
We now prove this lemma used in the proof.
Let for each i ∈ [1, K], G n i (i, f ) be the functional empirical process based We consider the three terms in (6.20), that is the C(n, i), 1 ≤ i ≤ 3, defined in (6.5), (6.10) and in (6.11), and prove that each of them converges to a random variable C(i) depending on the limiting Gaussian processes G(i, ·) of G n i (i, ·). This is enough to prove the asymptotic normality. The variance ϑ 2 1 will be nothing else but that of C(1) + C(2) + C(3). Firstly, we treat C(n, 1). Remark that conditionally on (n * = n), the random sequences {Y i,j , 1 ≤ i ≤ n i , 1 ≤ i ≤ K} are independent and only depend on the ω 2 ∈ Ω 2 . We have Then, by (6.5) and replacing n * i by n i , i = 1, ..., K, we get This implies that We finally have that Since the G i, (g − g i ) G −1 i are independent, centered and Gaussian, we get that In the sequel we take g 0 (x) = g 0 (x) × e(x) and ν 0 (x) = ν 0 (x) × e(x), and (g 0 , ν 0 ) ∈ (g, g 1 , ..., g K ) × (ν, ν 1 , ..., ν K ) and i = 1, ..., K. Then we arrive Secondly, one has )ds, and thus Finally, one has s)))ds. We remember that ν is of the form where ν a is continuous on compact sets [0, L], L > 0. Since, as n → ∞, sup s∈(0,1) |V n i (i, s) − s| → 0, a.s, we see that, for large values of n, theses integrals are performed at most on some interval [0, G i (Z) + ε], which includes those s satisfying V n i (i, s) ≤ G i (Z). By the assumptions, the functions ν a and G are continuous on such compact sets. Thus We surely have, by continuity of G h on 0, G −1 i (G(Z) + ε) , sup We obtain here a continuous modulus of the uniform empirical process (see Shorrack and wellner [20], page 531) and then a n log a n ).
We finally get R n = O −a n log a n 1 0 ν(G −1 i (s))ds → 0 and we arrive at (6.23) We are now going to compute the variance ϑ 2 1 based on the independent functional Browian bridges G(i, ·) which are limits of the functional empirical process G n (i, ·) respectively associated with {G i (Y i,j ), 1 ≤ i ≤ n i }, i = 1, .., K. Straightforward calculations give what comes. First We denote l i = (g − g i )G −1 i in the sequel for sake of simplicity. Next for we have we have Now by using the independence of the centered stochastic process G(h, · · · ) for differents values of h ∈ {1, ..., K}, one gets

and then
We have next

Conclusion
We just illustrated how apply our results for the Sen Measure and the Senegalese database ESAM I and the Mauritanian EPCV 2004 data. But It would be more interesting and instructive to conduct large scale data-driven for the West African databases for example, for several measures. It would also be interesting to see the influence of the Kakwani parameter k on the results. This study is underway.

Appendix
We would like to provide indications to the reader for using the techniques developped here. We have a zipped file at : http : //www/uf rsat.org/lerstad/sen − decomposabilite.rar It includes the executable sendecomp.exe file which performs the computation of dg. Here is how to proceed : (i) Download the zipped file and unzip him in a folder named, for instance, sen-decomposabilite. (ii) Upload in the sen-decomposabilite folder the following user files : The income file dep.txt of size n at most equal to 10000, the equivalentadult file eq.txt of the same size n and finally the labels file labels.txt including the names of the different strates. If the income file is already scaled for individuals, use an eq.txt file of size n having unity at each line. Le nomber of labels is at most equal to 15. They must be enumarated from to 1 to KK < 16. (iii) Execute sendecomp.exe by clicking on it. The user is prompted to provide the income file name, the equivalen-adult file name and the labels file name without the suffixs .txt. (v) The package provides the sen measures value for the differents strates and report the gap of decomposability value. (vi) For the user's practice we provided in the zipped folder the following income variables (depm.txt), equivalent-adult variable (eom.txt) and labels (here areas) file named after regm.txt. (vi) If the data size exceeds n = 10000 or the strates number exceeds KK = 15, the user is free to write to the authors and adapted packages will be provided.
Finally for those who want to set their own packages in some langage, we provide a Visual Basic module including the main program and the subroutines.