Multidimensional p-p plots and precedence tests for point processes on R d

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 precedence test and the corresponding p-p plot that is appropriate for a test of F = G against the alternative that F ≺ G for various stochastic orders on ℝ
 d
 . The test statistic is consistent and 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 Kendall’s process, and in one case the resulting test statistic has the novelty of being distribution free.

One-dimensional p-p plots and precedence tests for point processes on ℝ d

Let Xi,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 lineto 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 quantiles 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 !ft, are valid for 3?-valued data.

/pdf/p-p-plots-and-precedence-tests-for-planar-point-processes-3u05v6hft8.pdf

Adriana Jordan

Papers

p-p plots and precedence tests for planar point processes