Goodness of Fit Tests in Models for Life History Data Based on Cumulative Hazard Rates

Abstract

To check the validity of an assumed parametric model for survival data, one may compare $\hat{A}(t)$, the nonparametric Nelson-Aalen plot of the cumulative hazard rate, with $A(t, \hat{\theta})$, the estimated parametric cumulative hazard rate, $\hat{\theta}$ being for example the maximum likelihood estimator. Convergence in distribution of $\sqrt n (\hat{A}(t) - A(t, \hat{\theta}))$ and more general processes is studied in the present paper, employing the general framework of counting processes, which allows for quite general models for life history data and for quite general censoring schemes. The results are applied to the construction of $\chi^2$-type statistics for goodness of fit. Cramer-von Mises and Kolmogorov-Smirnov type tests are presented in the case where the unknown parameter is one-dimensional. Power considerations are also included, and some optimality results are reached. Finally tests are constructed for the hypothesis that the unspecified hazard rate part in Cox's regression model follows a parametric form.