A robust and efficient estimator for the tail index of inverse Pareto distribution

Abstract

Based on the probability integral transform statistic, Finkelstein et al. (2006) have proposed a robust estimator for the shape parameter of Pareto distribution. In this paper, based on the same method, a robust and efficient estimator for the shape parameter of the inverse Pareto distribution is developed assuming that the threshold parameter is known. To study the robustness properties of this new estimator, we derive the asymptotic variance, breakdown point and gross error sensitivity. However, since the inverse Pareto distribution is literally an inverse of the Pareto distribution, some derivations and proofs for the probability integral transform statistic estimator presented in this paper are found closely related to those provided by Finkelstein et al. (2006). The performance of this new estimator and the maximum likelihood estimator is assessed through a simulation study. For the application, an inverse Pareto distribution is fitted to the lower tail data of Malaysian household incomes for the year of 2014, involving the proposed estimator in order to allow for the presence of outliers. Based on the inverse Pareto model, the parametric Lorenz curve is fitted and the Gini coefficient is estimated to measure the income inequality of poor households in Malaysia.