###
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}

^{1} *Department of Public Health, University of Western Australia,*

Nedlands, WA 6907, Australia

^{2} *Department of Mathematics, University of Western Australia,*

Nedlands, WA 6907, Australia

^{3} *Department 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.

**Source**
( TeX ,
DVI ,
PS )