A generalized approach to the two sample problem : the quantile approach

Abstract

Parzen (1979) suggests a location and scale model for the quantile function (inverse distribution function) of a random variable. We extend this model to the two sample and k-sample problems and some results are given which, when fully implemented, will yield more general solutions in the analysis of variance. Most of the work here concerns the location and scale model suggested by Parzen (1980) for the two sample problem for testing the equality of two distribution functions versus local alternatives. We implement this model (its tests and estimators) for seven underlying densities. We then provide criteria for choosing or determining whether an underlying density models the differences of the two samples adequately. These criteria allow one to choose the best of several underlying densities for the data. We illustrate these techniques by analyzing data sets from the literature and making comparisons with other authors' techniques. We also show how the Parzen (1980) model is related to many of the techniques developed for studying differences of two samples over the past 50 years. We suggest extensions of Parzen's model. Finally, we give a few simulated examples and suggest what type of simulation study is needed to further define the usefulness of the various models presented in the dissertation.