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Kernel density and hazard function estimation in the presence of censoring. (English) Zbl 0661.62028
The authors study nonparametric estimation of a density f \((=F')\) and failure rate \(\lambda =f/1-F\) based on an i.i.d. sample \(X_ 1,...,X_ n\) with density f when the observations are at risk of being censored from the right by an i.i.d. sequence \(Y_ 1,...,Y_ n\) with density g \((=G')\). The estimator of f studied here is the kernel estimator \[ f_ n(t)=h_ n^{-1}\int^{\infty}_{-\infty}K((t-x)/h_ n)\hat F_ n(dx), \] where \(\{h_ n\}\) is a sequence of bandwidths tending to zero, K is a smooth probability kernel and 1-\^F\({}_ n(x)\) is the Kaplan- Meier estimator of the survival function 1-F(x).
An almost sure representation and an in probability representation of \(f_ n\) are obtained in terms of a sum of independent random variables plus negligible remainder terms under the conditions that the kernel K(.) is continuously differentiable and vanishing outside some finite interval.
As a consequence of these representations, results on the rate of pointwise convergence, rate of uniform convergence, asymptotic distribution and limit distribution for the maximal deviation of \(f_ n\) from \(\bar f_ n\) are derived where \[ \bar f_ n(t)=h_ n^{- 1}\int^{\infty}_{-\infty}K((t-x)/h_ n)F(dx). \] Similar results were obtained for the failure rate \(\lambda\). The results obtained here improve earlier work by A. Foeldes, L. Rejtoe and B. B. Winter [Period. Math. Hung. 12, 15-29 (1981; Zbl 0461.62038)], H. Ramlau-Hansen [Ann. Stat. 11, 453-466 (1983; Zbl 0514.62050)], M. A. Tanner and W. H. Wong [ibid., 989-993 (1983; Zbl 0546.62017)], B. S. Yandell [ibid., 1119-1135 (1983; Zbl 0598.62050)] and others.
Reviewer: B.L.S.Prakasa Rao

MSC:
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
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