Nonparametric estimation and consistency for renewal processes

Abstract In reliability or medical studies, we may only observe each ongoing renewal process for a certain period of time. When the underlying distribution F is arithmetic, Vardi (Ann. Statist. 10 (1982b), 772–785) developed the RT algorithm for nonparametric estimation. In this paper we extend the study to the nonarithmetic case and show that the choice of an arbitrary constant in the RT algorithm can be avoided. We also prove the strong consistency of the maximum likelihood estimators of the mean of F and F restricted to the interval [0, b), if the lengths of the observation periods converge to b.

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