Asymptotic properties of a class of mixture models for failure data

The interior and boundary cases

H. T V Vu*, R. A. Maller, X. Zhou

*Corresponding author for this work

Research output: Contribution to journalArticle

6 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)627-653
Number of pages27
JournalAnnals of the Institute of Statistical Mathematics
Volume50
Issue number4
Publication statusPublished - 1998
Externally publishedYes

Keywords

  • Boundary hypothesis tests
  • Censored survival data
  • Covariates
  • Exponential family
  • Failure time data
  • Immune proportion
  • Mixture models

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