Multidimensional p–p plots and precedence tests for point processes on ℜd

Given two univariate distributions F and G, the hypothesis that F=G can be tested against the alternative that F<G using a precedence test statistic, or more generally the procentile–procentile plot of G against F. In this paper we propose a d-dimensional generalization of the p–p plot that can be used to construct a precedence test of F=G against the alternative that F≺G for various stochastic orders on ℜd. The p–p plot process is shown to converge to a Gaussian limit. Under the null hypothesis, the p–p plot can be regarded as a two sample version of the empirical copula process. In one dimension, the test statistics proposed correspond to classical nonparametric rank-based tests.