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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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##### References:
 [1] Bickel, P.J; Rosenblatt, M, On some global measures of the deviations of density function estimates, Ann. statist., 1, 1071-1095, (1973) · Zbl 0275.62033 [2] Blum, J.R; Susarla, V, Maximal deviation theory of density and failure rate function estimates based on censored data, (), 213-222 · Zbl 0462.62030 [3] Breslow, N; Crowley, J, A large sample study of the life table and product limit estimates under random censorship, Ann. statist., 2, 437-459, (1974) · Zbl 0283.62023 [4] Csörgö, S; Horvath, L, The rate of strong uniform consistency for the product-limit estimator, Z. wahrsch. verw. gebiete, 62, 411-426, (1983) · Zbl 0488.60043 [5] Dvoretzky, A; Kiefer, J; Wolfowitz, J, Asymptotic minimax character of the samle distribution function and of the classical multinomial estimator, Ann. math. statist., 27, 642-669, (1956) · Zbl 0073.14603 [6] Falk, M, Kernel estimation of a density in an unknown endpoint of its support, South african statist. J., 18, 91-96, (1984) · Zbl 0544.62042 [7] Földes, A; Rejtö, L; Winter, B.B, Strong consistency properties of nonparametric estimators for randomly censored data. II. estimation of density and failure rate, Period. math. hungar., 12, 15-29, (1981) · Zbl 0461.62038 [8] Hall, P, Laws of the iterated logarithm for nonparametric density estimators, Z. wahrsch. verw. gebiete, 56, 47-61, (1981) · Zbl 0443.62027 [9] Kaplan, E.L; Meier, P, Nonparametric estimation from incomplete observations, J. amer. statist. assoc., 53, 457-481, (1958) · Zbl 0089.14801 [10] Mielniczuk, J, Some asymptotic properties of kernel estimators of a density function in case of censored data, Ann. statist., 14, 766-773, (1986) · Zbl 0603.62047 [11] Nelson, W, Theory and applications of hazard plotting for censored failure data, Technometrics, 14, 945-966, (1972) [12] Padgett, W.J; Mc Nichols, D.T, Nonparametric density estimation from censored data, Comm. statist. A—theory methods, 13, 1581-1611, (1984) · Zbl 0552.62021 [13] Parzen, E, On estimation of a probability density function and mode, Ann. math. statist., 33, 1065-1076, (1962) · Zbl 0116.11302 [14] Ramlau-Hansen, H, Smoothing couting process intensities by means of kernel functions, Ann. statist., 11, 453-466, (1983) · Zbl 0514.62050 [15] Stute, W, A law of the logarithm for kernel density estimators, Ann. probab., 10, 414-422, (1982) · Zbl 0493.62040 [16] Tanner, M.A; Wong, W.H, The estimation of the hazard function from randomly censored data by the kernel method, Ann. statist., 11, 989-993, (1983) · Zbl 0546.62017 [17] Yandell, B.S, Nonparametric inference for rates with censored survival data, Ann. statist., 11, 1119-1135, (1983) · Zbl 0598.62050
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