Jackknife Empirical Likelihood Inference For The Pietra Ratio

Pietra ratio (Pietra index), also known as Robin Hood index, Schutz coefficient (RicciSchutz index) or half the relative mean deviation, is a good measure of statistical heterogeneity in the context of positive-valued data sets. In this thesis, two novel methods namely “adjusted jackknife empirical likelihood” and “extended jackknife empirical likelihood” are developed from the jackknife empirical likelihood method to obtain interval estimation of the Pietra ratio of a population. The performance of the two novel methods are compared with the jackknife empirical likelihood method, the normal approximation method and two bootstrap methods (the percentile bootstrap method and the bias corrected and accelerated bootstrap method). Simulation results indicate that under both symmetric and skewed distributions, especially when the sample is small, the extended jackknife empirical likelihood method gives the best performance among the six methods in terms of the coverage probabilities and interval lengths of the confidence interval of Pietra ratio; when the sample size is over 20, the adjusted jackknife empirical likelihood method performs better than the other methods, except the extended jackknife empirical likelihood method. Furthermore, several real data sets are used to illustrate the proposed methods. INDEX WORDS: Bootstrap method, Coverage probability, Jackknife empirical likelihood, Adjusted jackknife empirical likelihood, Extended jackknife empirical likelihood JACKKNIFE EMPIRICAL LIKELIHOOD INFERENCE FOR THE PIETRA RATIO