The asymptotic theory of the poverty intensity in view of extreme value theory for two simple cases

Abstract

Let $Y_1,Y_2\dots$ be independent observations of the income variable of some given population, with underlying distribution $G$. Given a poverty line $Z$, then for each $n\geq 1$, $q=q_n$ is the number of poor in the population. The general form of poverty measures used by economists to monitor the welfare evolution of this population is $$P_n=\frac{1}{a(q)b(n)}\sum^q_{j=1}c(n,q,j)d\left(\frac{Z-Y_{j,n}}{Z}\right).$$ This class includes the most popular poverty measures like the Sen, Shorrocks and Greer-Foster-Thorbecke statistics. We give a complete asymptotic normality theory in the framework of extreme value theory. In this paper, the poverty intensity is studied in two simple cases: Pareto and exponential distributions. Simulations and applications to the Senegalese data are given.