Abstract
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.
Highlights
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
First we introduce some basics elements: let N be a statistical universe of individuals, 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of them it is possible to determine its level of income following e.g. [12, 17], for a random sample of n individuals withdrawn from this population, a measure or classic index of poverty is a function P : Rn++1 → [0, 1], where the value of P(y, z) indicates the degree or level of poverty associated with the vector of incomes y = (y1, y2, . . . , yn) ∈ Rn+ and the fixed poverty line z ∈ R+, such that any j-th individual of the random sample is considered poor if yj < z
With the research published by Sen in 1976 about the first axioms or properties of the axiomatic method of poverty, 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
Summary
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.
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