ASYMPTOTIC PROPERTIES OF A CLASS OF MIXTURE MODELS FOR FAILURE DATA: THE INTERIOR AND BOUNDARY CASES

H. T. V. VU^{1}, R. A. MALLER^{2} AND X. ZHOU^{3}

1Department of Public Health, University of Western Australia, Nedlands, WA 6907, Australia 2Department of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia

3Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

(Received February 17, 1997; revised September 30, 1997)

Abstract. We analyse an exponential family of distributions which generalises the exponential distribution for censored failure time data, analogous to the way in which the class of generalised linear models generalises the normal distribution. The parameter of the distribution depends on a linear combination of covariates via a possibly nonlinear link function, and we allow another level of heterogeneity: the data may contain “immune” individuals who are not subject to failure. Thus the data is modelled by a mixture of a distribution from the exponential family and a “mass at infinity” representing individuals who never fail. Our results include large sample distributions for parameter estimators and for hypothesis test statistics obtained by maximising the likelihood of a sample. The asymptotic distribution of the likelihood ratio test statistic for the hypothesis that there are no immunes present in the population is shown to be “non-standard”; it is a 50-50 mixture of a chi-squared distribution on 1 degree of freedom and a point mass at 0. Our analysis clearly shows how “negligibility” of individual covariate values and “sufficient followup” conditions are required for the asymptotic properties.

Key words and phrases: Censored survival data, immune proportion, covariates, mixture models, failure time data, exponential family, boundary hypothesis tests.