Bernstein polynomial model for nonparametric multivariate density

ABSTRACT

In this paper, we study the Bernstein polynomial model for estimating the multivariate distribution functions and densities with bounded support. As a mixture model of multivariate beta distributions, the maximum (approximate) likelihood estimate can be obtained using EM algorithm. A change-point method of choosing optimal degrees of the proposed Bernstein polynomial model is presented. Under some conditions, the optimal rate of convergence in the mean ${\chi }^2$-divergence of new density estimator is shown to be nearly parametric. The method is illustrated by an application to a real data set. Finite sample performance of the proposed method is also investigated by simulation study and is shown to be much better than the kernel density estimate but close to the parametric ones.

KEYWORDS:

• Approximate Bernstein polynomial model
• beta mixture
• maximum likelihood
• multivariate density estimation
• nonparametric model

Acknowledgments

The authors are grateful to the Editor and two referees for their useful comments some of which really helped to improve upon our original submission.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Tao Wang is supported by the Natural Science Foundation of Heilongjiang Province, China (Grant No. A2017006).