Some nonparametric methods in estimating the hazard rate function



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Texas Tech University


The hazard rate function h{x) = f{x)/[1-F{x)], corresponding to a distribution function F with density function / , is one of the most important parameters in reliability and other fields, since h{x)dx can be interpreted as the probability that an object fails in the time interval [x,x + dx] given that the object has survived to time x. A problem of considerable interest, especially to reusability engineers, is the estimation of h from a sample of n independent and identically distributed nonnegative lifetimes X1 , X2,…,Xn , with or without censoring.

By censoring we mean that the occurrence of the random event of interest (called a failure) is prevented by the previous occurrence of another event (called a censoring event). If the censoring event is also a random variable, we have the randomly censored model. If there are several randomly censoring events, we have the multiple competing risks model.

This paper discusses three nonparametric estimators of the hazard rate function h: the Fourier integral estimator, the nearest neighbor estimator in the multiple competing risks model, and the discrete maximum penalized-likelihood estimator in the randomly censored model. The asymptotic behavior of these estimators will be studied and conditions for strong consistency will be given.