Statistical analysis of poverty and inequality measures

Despite the growing interest in poverty and inequality studies and the large standard errors found in many empirical studies, most of the work in this area neglects statistical inference. Two types of inference procedures for poverty and inequality measures have been considered: asymptotic distributions and bootstrapping. These methods can be quite unreliable, even with fairly large samples, but no study has proposed provably valid exact inference procedures for such problems. We propose such ones.;In the second and third papers, we develop such inference methods for the Foster, Greer and Thorbecke (FGT, 1984) poverty measures (paper 2) and the most popular inequality measures (paper 3): the generalized entropy measures, the Theil index, the Lorenz curve, the Gini index, the Atkinson measures, the mean logarithmic deviation, and the logarithmic variation. We first observe that these poverty and inequality indicators can be interpreted as functions of the expectations of random variables which are themselves functional of distribution functions, where the involved variables can be either bounded or unbounded. Using projection techniques, we then derive finite-sample nonparametric confidence intervals for the mean of a bounded random variable from confidence bands for the distribution of the underlying variable. When the random variable is unbounded, we propose a generalized projection principle for distribution functions which tails are bounded by a Pareto distribution. Then, we apply these procedures to the FGT poverty measures and to inequality measures.;Monte Carlo simulations are performed in the three papers to study the relative performance of the inference methods and illustrate how to choose the regularization parameter. The results show that the regularized statistics yield more powerful goodness-of-fit tests than the existing ones when applied to distributions with more discrepancy in the tails. Likewise, the CBs for distribution functions and the confidence intervals based on these regularized statistics have a better performance. The simulations show that asymptotic and bootstrap confidence intervals for the mean can fail to provide reliable inference, while the proposed methods are robust and yield shorter confidence intervals. As an illustration, we analyze the profile of poverty and inequality of Mexico in 1998 using households' survey data (papers 2 and 3). The results show that the widths of the asymptotic confidence intervals are often too small to be realistic while those of the bootstrap can be 10 times larger than the widths delivered by exact methods. The study shows that the poverty profile of Mexican households depends greatly on the type of households' head: poverty levels and inequality among households with a male head or an educated head are much smaller than those among other households. Hence, policies aimed at reducing illiteracy and at securing the income of households with a female head could help reduce poverty and inequality in rural Mexico.;Keywords: nonparametric inference; Kolmogorov-Smirnov; Anderson-Darling; Eicker; empirical distribution; mean; poverty; inequality; regularization; Paretian heavy tail. JEL codes: C01, C12, C14, O11.;In the first paper, we build nonparametric confidence bands for distribution functions by inverting goodness-of-fit tests based on improved standardized Kolmogorov-Smirnov statistics (KS, henceforth). Despite its popularity, the KS test does not allow to discriminate a lot between distributions that differ mostly through their tails. To correct this drawback, weighted KS statistics based on the three common principles in econometrics (the Wald, Lagrange multiplier, and likelihood-ratio principles) are proposed respectively by Anderson and Darling (1952), Eicker (1979), and Berk and Jones (1979). However, they also suffer from drawbacks because standard errors can be very close to zero. To correct these, we propose improved weighted KS statistics obtained by adding a regularization term in the denominator of the Anderson-Darling and the Eicker statistics and derive from them exact nonparametric confidence bands for distribution functions. We show that in the continuous case, these confidence bands are independent of the distribution assumed under the null hypothesis and are conservative for noncontinuous distributions. In the noncontinuous case, we derive monotonicity properties to narrow the confidence bands without altering their reliability.