Goodness of fit tests in models for life history data based on cumulative hazard rates. (English) Zbl 0714.62037

Hjort, Nils Lid

Ann. Stat. 18, No. 3, 1221-1258 (1990).

Author’s summary: 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, 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. Cramér-von Mises and Kolmogorv-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.

Reviewer: S.M.Shkol’nik

MSC:

62G10	Nonparametric hypothesis testing
60F05	Central limit and other weak theorems
62N05	Reliability and life testing

Keywords:

chi-squared statistics; Cramer-von Mises statistic; survival data; nonparametric Nelson-Aalen plot of the cumulative hazard rate; estimated parametric cumulative hazard rate; maximum likelihood estimator; Convergence in distribution; counting processes; life history data; censoring; goodness of fit; Kolmogorv-Smirnov type tests; Power considerations; optimality results; Cox’s regression model