A bayesian nonparametric approach for the two-sample problem

Abstract

In this work, we discuss the so-called two-sample problem (PEARSON; NEYMAN, 1930) assuming a nonparametric Bayesian approach. Considering X 1 ,...,X n and Y 1 ,...,Y m two inde- pendent i.i.d samples generated from P 1 and P 2 , respectively, the two-sample problem consists in deciding if P 1 and P 2 are equal. Assuming a nonparametric prior, we propose an evidence index for the null hypothesis H 0 : P 1 = P 2 based on the posterior distribution of the distance d(P 1 ,P 2 ) between P 1 and P 2 . This evidence index has easy computation, intuitive interpretation and can also be justified in the Bayesian decision-theoretic context. Further, in a Monte Carlo simulation study, our method presented good performance when compared to the well known Kolmogorov-Smirnov test, the Wilcoxon test as well as a recent testing procedure based on Polya tree process proposed by Holmes (HOLMES et al., 2015). Finally, we applied our method to a data set about scale measurements of three different groups of patients submitted to a questionnaire for Alzheimer’s disease diagnostic.