A histogram estimator of the hazard rate with censored data
Let \(X_ 1,...,X_ n\) denote lifetimes for the n subjects under study, and \(C_ 1,...,C_ n\) be the corresponding censoring times. The observed random variables are the \(Z_ i\) and \(\delta_ i\) where \(Z_ i=\min (X_ i,C_ i)\) and \(\delta_ i=I(X_ i\leq C_ i)\). The authors assume throughout the paper that (i) \(X_ 1,...,X_ n\) are nonnegative and i.i.d. with common d.f. F and continuous density f, (ii) \(C_ 1,...,C_ n\) are nonnegative and i.i.d. with common continuous d.f. G and continuous density g, and (iii) lifetimes and censoring times are independent. The problem considered here is estimation of the hazard rate function given by \[ \lambda (x)=f(x)/(1-F(x)),\quad F(x)<1. \] The proposed estimator is based on random spacings of the order statistics of uncensored observations. The authors establish pointwise large sample properties of the estimator including strong consistency on a bounded interval, and two asymptotic results.
scientific article; zbMATH DE number 3990636