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A histogram estimator of the hazard rate with censored data. (English) Zbl 0612.62059
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.
Reviewer: V.D.Konakov

##### MSC:
 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 62P10 Applications of statistics to biology and medical sciences; meta analysis
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