Nonparametric Estimation of a Vector-Valued Bivariate Failure Rate

Abstract

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.