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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(β 1 z 1 ++β p z p )λ 0 (t),

where λ 0 (·) is an arbitrary unknown function, β 1 ,,β p are unknown parameters and (z 1 ,,z p ) is the vector of explanatory variables. A modified likelihood function is obtained for inference about β 1 ,,β 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:
62J05Linear regression
62N05Reliability and life testing (survival analysis)
62F10Point estimation