A bivariate histogram density estimator: Consistency and asymptotic normality

This paper presents a nonparametric histogram density estimator based on the spacings of order statistics. This estimator generalizes to the bivariate case the univariate histogram estimator proposed by Van Ryzin (1973). The first of the two theorems in this paper gives conditions under which the estimator is pointwise strongly consistent. The second theorem provides conditions for the asymptotic normality of the estimator for points at which the density function possesses continuous partial derivatives of second order.