The Annals of Statistics

Asymptotic theory for the correlated gamma-frailty model

Erik Parner

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The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.

Article information

Ann. Statist., Volume 26, Number 1 (1998), 183-214.

First available in Project Euclid: 28 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M09: Non-Markovian processes: estimation
Secondary: 62G05: Estimation

Nonparametric maximum likelihood estimation survival data heterogeneity correlated frailty central limit theorem semiparametric models


Parner, Erik. Asymptotic theory for the correlated gamma-frailty model. Ann. Statist. 26 (1998), no. 1, 183--214. doi:10.1214/aos/1030563982.

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