A Khmaladze-transformed test of fit with ML estimation in the presence of recurrent events

Abstract

This article provides a goodness-of-fit test for the distribution function or the survival function in a recurrent event setting, when the inter-event time parametric structure $F(·;\mathit{\theta })$ is estimated from the observed data. Of concern is the null hypothesis that the inter-event time distribution is absolutely continuous and belongs to a parametric family $\mathfrak{F}=\{F(·;\mathit{\theta }):\mathit{\theta }\in \Theta \subseteq {\mathfrak{R}}^q\}$, where the q-dimensional parameter space is neither known nor specified. We proposed a Khmaladze martingale-transformed type of test (Khmaladze, 1981), adapted to recurrent events. The test statistic combines two likelihood sources of estimation to form a parametric empirical process: (1) a product-limit nonparametric maximum likelihood estimator (NPMLE; Peña et al., 2001a) that is a consistent estimator of F, $\widehat{F}$ say, and (2) a point process likelihood estimator $F(·;\widehat{\mathit{\theta }})$ (Jacod, 1974/1975). These estimators are combined to construct a Kolmogorov-Smirnov (KS) type of test (Kolmogorov 1933; Smirnov, 1933). Empirical process and martingale weak convergence frameworks are utilized for theoretical derivations and motivational justification of the proposed transformation. A simulation study is conducted for performance assessment, and the test is applied to a problem investigated by Proschan (1963) on air-conditioning failure in a fleet of Boeing 720 jets.

Keywords:

• Empirical process
• finite elements discretization
• Khmaladze transform
• KS test
• martingales
• recurrent event

SUBJECT CLASSIFICATIONS:

• 62L05
• 60G42
• 60G44
• 62F12
• 62N05

Acknowledgments

The authors are indebted to the Editor, the editorial staff, and the reviewers for their suggestions and constructive criticism, which led to a much improved version of the article.

Additional information

Funding

Part of the methodological development has been supported by the NCI and NIH Specialized Program of Research Excellence in Neuroendocrine Tumors Grant No. 5 P50 CA174521-03.