Statistica Sinica 17(2007), 1023-1046

A PROFILE LIKELIHOOD THEORY FOR THE

CORRELATED GAMMA-FRAILTY MODEL

WITH CURRENT STATUS FAMILY DATA

I-Shou Chang$^1$, Chi-Chung Wen$^2$ and Yuh-Jenn Wu$^3$

$^1$National Health Research Institutes, $^2$Tamkang University

and $^3$Chung Yuan Christian University

Abstract: A profile likelihood inference is made for the regression coefficient and frailty parameters in the correlated gamma-frailty model for current status family data. With the introduction of an identifiability assumption, the identifiability of the parameters and the existence of the nonparametric maximum likelihood estimate (NPMLE) are established, the consistency and convergence rate of the NPMLE are obtained, the invertibility of the efficient Fisher information matrix is proved, and a quadratic expansion of the profile likelihood is established. From these, we show that the NPMLE of the parameters of interest is asymptotically normal and efficient, its covariance matrix can be estimated consistently by means of the profile likelihood, and the likelihood ratio test is asymptotically chi-squared. A simulation study is carried out to illustrate the numerical performance of the likelihood ratio test.

Key words and phrases: Current status family data, least favorable submodel, likelihood ratio statistic, nonparametric maximum likelihood estimate, profile likelihood.