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.
All Science Journal Classification (ASJC) codes
- Statistics, Probability and Uncertainty
- Statistics and Probability