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

• Published in: Sequential analysis Vol. 38; no. 3; pp. 318 - 341
• Main Authors: Zamba, K. D., Adekpedjou, Akim
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.07.2019; Taylor & Francis Ltd
• Subjects: Air conditioners; Air conditioning; Aircraft industry; Distribution functions; Economic models; Empirical process; finite elements discretization; Goodness of fit; Khmaladze transform; KS test; Martingales; Maximum likelihood estimation; Maximum likelihood estimators; Null hypothesis; Performance assessment; recurrent event; Statistical tests
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2019.1648920

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 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 , 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, say, and (2) a point process likelihood estimator (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.