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September, 1977 Nonparametric Estimation of a Vector-Valued Bivariate Failure Rate
Ibrahim A. Ahmad, Pi-Erh Lin
Ann. Statist. 5(5): 1027-1038 (September, 1977). DOI: 10.1214/aos/1176343957


Let $\mathbf{X} = (X_1, X_2)'$ be a bivariate random vector distributed according to an absolutely continuous distribution function $F(\mathbf{x})$ which has first partial derivatives. Let $\bar{F}(\mathbf{x}) = P(X_1 > x_1, X_2 > x_2).$ The vector-valued bivariate failure rate is defined as $\mathbf{r}(\mathbf{x}) = (r_1(\mathbf{x}), r_2(\mathbf{x}))',$ where $r_i(\mathbf{x}) = -\partial \ln \bar{F}(\mathbf{x})/\partial x_i (i = 1, 2)$. In this paper, we propose a smooth nonparametric estimate $\hat\mathbf{r}(\mathbf{x})$ of $\mathbf{r}(\mathbf{x})$ using Cacoullos' (Ann. Inst. Statist. Math. 18 (1966), 181-190) multivariate density estimate. Regularity conditions are obtained under which $\hat\mathbf{r}(\mathbf{x})$ is shown to be pointwise strongly consistent. A set of sufficient conditions is given for the strong uniform consistency of $\hat\mathbf{r}(\mathbf{x})$ over a subset $S$ of $R^2$ where $\bar{F}(\mathbf{x})$ is bounded below by $\varepsilon > 0.$ The joint asymptotic normality of the estimate evaluated at $q$ distinct continuity points of the failure rate is established. The methods and results presented in this paper can be generalized to any finite dimensional case in a straightforward manner.


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Ibrahim A. Ahmad. Pi-Erh Lin. "Nonparametric Estimation of a Vector-Valued Bivariate Failure Rate." Ann. Statist. 5 (5) 1027 - 1038, September, 1977.


Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0369.62041
MathSciNet: MR451517
Digital Object Identifier: 10.1214/aos/1176343957

Primary: 62G05
Secondary: 62E20 , 62G20 , 62N05

Keywords: and strong univorm consistency , Bernstein's inequality , Cacoullos' multivariate density estimate , hazard gradient , kernel function , limiting distribution of the estimate , strong consistency

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 5 • September, 1977
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