We consider a class of semiparametric regression models which areone-parameter extensions of the Cox [J. Roy. Statist. Soc. Ser. B 34 (1972)187-220] model for right-censored univariate failure times. These models assumethat the hazard given the covariates and a random frailty unique to eachindividual has the proportional hazards form multiplied by the frailty. The frailty is assumed to have mean 1 within a known one-parameter family ofdistributions. Inference is based on a nonparametric likelihood. The behaviorof the likelihood maximizer is studied under general conditions where thefitted model may be misspecified. The joint estimator of the regression andfrailty parameters as well as the baseline hazard is shown to be uniformlyconsistent for the pseudo-value maximizing the asymptotic limit of thelikelihood. Appropriately standardized, the estimator converges weakly to aGaussian process. When the model is correctly specified, the procedure issemiparametric efficient, achieving the semiparametric information bound forall parameter components. It is also proved that the bootstrap gives validinferences for all parameters, even under misspecification. We demonstrate analytically the importance of the robust inference in severalexamples. In a randomized clinical trial, a valid test of the treatment effectis possible when other prognostic factors and the frailty distribution are bothmisspecified. Under certain conditions on the covariates, the ratios of theregression parameters are still identifiable. The practical utility of theprocedure is illustrated on a non-Hodgkin's lymphoma dataset.
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