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Some asymptotic properties of kernel estimators of a density function in case of censored data. (English) Zbl 0603.62047
The author considers a random censorship model based on nonnegative independent random variables X and Y with densities f and g resp.. Given a sample $$Z_ i=\min (X_ i,Y_ i)$$, $$\delta_ i=\chi (X_ i\leq Y_ i)$$, $$1\leq i\leq n$$, he investigates the following density estimator for $$f$$: $\hat f_ n(x)=(1/h)\int_{{\mathbb{R}}}K((x-y)/h)dF_ n^{(KM)}(y),$ where K is a density itself and $$F_ n^{(KM)}$$ is the Kaplan-Meier estimator for $$F_ X$$. [See also J. Blum and V. Susarla, Multivariate analysis V, Proc. 5th int. Symp., Pittsburgh 1978, 213-222 (1980; Zbl 0462.62030) or A. Földes, L. Rejtö and B. B. Winter, Period. Math. Hung. 12, 15-29 (1981; Zbl 0461.62038).]
Four types of bandwidths h are considered, a deterministic sequence $$h_ n$$, respectively a sequence $$h_{n_ 1}$$, where $$n_ 1=\#$$ of uncensored data, and N.N.-bandwidths $$R(n)$$, $$R(n_ 1)$$ resp. It is shown that in all cases $$\hat f_ n$$ can be approximated by kernel density estimators based on appropriate empirical distributions and the well known limit theorems for those apply and result in e.g. strong uniform consistency and asymptotic normality of $$\hat f_ n$$ (in Cor. (2,ii) read $$h_ n=o(n^{-1/5})$$ instead of $$O(n^{-1/3}))$$. It turns out that the variances of the N.N.-estimates are censoring free.