On the Discrete Cram\'er-von Mises Statistics under Random Censorship

TL;DR

This paper introduces nonparametric Cramér-von Mises tests for homogeneity of discrete variables under right-censoring, extending existing methods to handle infinitely many categories with complex asymptotic distributions.

Contribution

It develops weighted log-rank statistics for discrete models under censoring, providing a new test statistic with asymptotic distribution characterized as a series of chi-squared variables.

Findings

The test is consistent against all alternatives.

It accommodates crossing hazard functions.

Simulation results support theoretical properties.

Abstract

In this work, nonparametric log-rank-type statistical tests are introduced in order to verify homogeneity of purely discrete variables subject to arbitrary right-censoring for infinitely many categories. In particular, the Cram\'er-von Mises test statistics for discrete models under censoring is established. In order to introduce the test, we develop the weighted log-rank statistics in a general multivariate discrete setup which complements previous fundamental results of Gill (1980) and Andersen et al. (1982). Due to the presence of persistent jumps over the unbounded set of categories, the asymptotic distribution of the test is not distribution-free. The statistical test for a large class of weighted processes is described as a weighted series of independent chi-squared variables whose weights can be consistently estimated and the associated limiting covariance operator can be infinite-dimensional. The test is consistent to any alternative hypothesis and, in particular, it allows us to deal with crossing hazard functions. We also provide a simulation study in order to illustrate the theoretical results.