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Regression models and life-tables. (English) Zbl 0243.62041

The statistical analysis is considered of observations on non-negative variables subject to censoring on the right. That is, for each individual we may observe either the value of the random variable $T$ or that $T$ exceeds some given value, not necessarily the same for all individuals. Such data arise commonly in medical, actuarial and industrial contexts. For simplicity, call $T$ a failure time. Further it is assumed that there is available for each individual a vector of explanatory variables which may influence $T$. Possible approaches to the analysis are reviewed. Primarily the paper deals with a model in which the age-specific failure rate (hazard function) has the form

$exp\left({\beta }_{1}{z}_{1}+\cdots +{\beta }_{p}{z}_{p}\right){\lambda }_{0}\left(t\right),$

where ${\lambda }_{0}\left(·\right)$ is an arbitrary unknown function, ${\beta }_{1},\cdots ,{\beta }_{p}$ are unknown parameters and $\left({z}_{1},\cdots ,{z}_{p}\right)$ is the vector of explanatory variables. A modified likelihood function is obtained for inference about ${\beta }_{1},\cdots ,{\beta }_{p}$ by arguing conditionally on the observed failure times. From this likelihood tests and confidence regions are obtained. In the special case of a two-sample problem with proportional hazards, the test of the null hypothesis of zero difference reduces to a generalization to censored data of the most efficient two-sample rank test for exponential distributions. A number of generalizations are considered and the relation with stochastic models discussed.

Discussion of the paper by 15 contributors is included togeher with the author’s reply.

Reviewer: D. R. Cox

##### MSC:
 62J05 Linear regression 62N05 Reliability and life testing (survival analysis) 62F10 Point estimation