Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors

This paper considers multivariate extension of smooth estimator of the distribution and density function based on Bernstein polynomials studied in Babu et al. [2002. Application of Bernstein polynomials for smooth estimation of a distribution and density function. J. Statist. Plann. Inference 105, 377-392]. Multivariate version of Bernstein polynomials for approximating a bounded and continuous function is considered and adapted for smooth estimation of a distribution function concentrated on the hypercube [0, 1]^d, d > 1. The smoothness of the resulting estimator, naturally lends itself in a smooth estimator of the corresponding density. The functions with other compact or non-compact support can be dealt through suitable transformations. The asymptotic properties, namely, strong consistency and asymptotic normality of the resulting estimators are investigated under α-mixing. This has been motivated by estimation of conditional densities in non-linear dynamical systems.