Searching the differences through the tails of distributions using an approach based on Mahalanobis distance and percentile bootstrap

Abstract

One of the main objectives of applied statistics is to determine whether two independent groups differ, and if so, to understand how they do. The Student t-test is the conventional method for doing this; however, it merely depends on a sensitive measure of location and a sampling distribution with some restrictive assumptions. There are some robust alternatives in the literature, but they concentrate on just one reference point over the entire distribution as well. It is often of interest to analyze the differences that occur in the tails of distributions, namely the quantiles. In this study, two independent groups are compared using a method based on a Mahalanobis distance and a percentile bootstrap approach. Two recently proposed and two well-known quantile estimators are used with the proposed method. Results for both theoretical distributions and real data sets are examined. Using the proposed method in conjunction with the Navruz and Özdemir (NO) estimator saved nominal Type-I error in all but five experimental cases out of 61 and it could potentially offer valuable insights into detecting the differences that occurred in the tails of the distributions.