# Nonparametric tests for transition probabilities in nonhomogeneous Markov processes

## TL;DR

This paper introduces nonparametric two-sample tests for comparing transition probabilities in nonhomogeneous Markov processes, with rigorous significance assessment and extensions to complex cases, validated through simulations and real data application.

## Contribution

It develops new nonparametric tests for transition probabilities in nonhomogeneous Markov processes, including extensions to complex scenarios and validation through simulations and real data.

## Key findings

- Tests perform well with small samples
- L2 and KS tests are consistent under fixed alternatives
- Application to breast cancer treatment data demonstrates practical utility

## Abstract

This paper proposes nonparametric two-sample tests for the direct comparison of the probabilities of a particular transition between states of a continuous time nonhomogeneous Markov process with a finite state space. The proposed tests are a linear nonparametric test, an L2-norm-based test and a Kolmogorov-Smirnov-type test. Significance level assessment is based on rigorous procedures, which are justified through the use of modern empirical process theory. Moreover, the L2-norm and the Kolmogorov-Smirnov-type tests are shown to be consistent for every fixed alternative hypothesis. The proposed tests are also extended to more complex situations such as cases with incompletely observed absorbing states and non-Markov processes. Simulation studies show that the test statistics perform well even with small sample sizes. Finally, the proposed tests are applied to data on the treatment of early breast cancer from the European Organization for Research and Treatment of Cancer (EORTC) trial 10854, under an illness-death model.