p-p plots and precedence tests for planar point processes

Abstract

Let X1, X2,..., Xn be a random sample of size n from a continuous distribution F and Y1 , Y2,...,Ym be a random sample of size m from a continuous distribution G. One of the ways to test the hypothesis of equality of F and G against the alternative that F < G when both distributions are univariate is to perform a precedence test -a test that not only requires only a portion of the samples, but which is distribution-free under the null hypothesis. The initial purpose of this thesis was to extend the notion of a precedence test to higher dimensions. In doing so, we found two different tests that are appropriate for both partial and complete data sets. These tests are based on two different extensions of the usual definition of a procentile-procentile plot -which is closely related to the precedence test statistic on the line- to the plane. The first of the above mentioned extensions involves the contours formed by the distribution function F; the second of our tests uses the marginal quartiles of F. For both extensions of the empirical p -- p plot, we have proven a Glivenko-Cantelli type of result. Also, we have developed their asymptotic convergence to Gaussian limits. The choice between tests based on these two plots depends on the kind of information that the data of our experiment generates. All the results presented here, although mostly presented for R2 , are valid for ℜd -valued data.