Nonparametric Estimation of a Vector-Valued Bivariate Failure Rate

Ahmad, Ibrahim A. ; Lin, Pi-Erh

Ann. Statist., Tome 5 (1977) no. 1, p. 1027-1038 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1977-09-14
Classification:  Bernstein's inequality,  Cacoullos' multivariate density estimate,  hazard gradient,  kernel function,  limiting distribution of the estimate,  strong consistency,  and strong univorm consistency,  62G05,  62G20,  62E20,  62N05