Mathematics and Statistics Vol. 12(2), pp. 142 - 156
DOI: 10.13189/ms.2024.120204
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Recursive Estimation of the Multidimensional Distribution Function Using Bernstein Polynomial

D. A. N. Njamen ^1,*, B. Baldagaï ¹, G. T. Nguefack ², A. Y. Nana ^3
1 Department of Mathematics and Computer Sciences, Faculty of Sciences, University of Maroua, Cameroon
² Department of Public Health, Faculty of Medicine and Biomedical Sciences, University of Yaoundé I, Cameroon
³ Laboratoire de Mathématiques et Informatique, Université de Douala, Cameroon

ABSTRACT

The recursive method known as the stochastic approximation method, can be used among other things, for constructing recursive nonparametric estimators. Its aim is to ease the updating of the estimator when moving from a sample of size n to n + 1. Some authors have used it to estimate the density and distribution functions, as well as univariate regression using Bernstein's polynomials. In this paper, we propose a nonparametric approach to the multidimensional recursive estimators of the distribution function using Bernstein's polynomial by the stochastic approximation method. We determine an asymptotic expression for the first two moments of our estimator of the distribution function, and then give some of its properties, such as first- and second-order moments, the bias, the mean square error (MSE), and the integrated mean square error (IMSE). We also determine the optimal choice of parameters for which the MSE is minimal. Numerical simulations are carried out and show that, under certain conditions, the estimator obtained converges to the usual laws and is faster than other methods in the case of distribution function. However, there is still a lot of work to be done on this issue. These include the studies of the convergence properties of the proposed estimator and also the estimation of the recursive regression function; the developments of a new estimator based on Bernstein polynomials of a regression function using the semi-recursive estimation method; and also a new recursive estimator of the distribution function, density and regression functions; when the variables are dependent.

KEYWORDS
Nonparametric Estimation, Stochastic Approximation Method, Multidimensional Distribution Function, Recursive Estimator, Multidimensional Bernstein Polynomial

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] D. A. N. Njamen , B. Baldagaï , G. T. Nguefack , A. Y. Nana , "Recursive Estimation of the Multidimensional Distribution Function Using Bernstein Polynomial," Mathematics and Statistics, Vol. 12, No. 2, pp. 142 - 156, 2024. DOI: 10.13189/ms.2024.120204.

(b). APA Format:
D. A. N. Njamen , B. Baldagaï , G. T. Nguefack , A. Y. Nana (2024). Recursive Estimation of the Multidimensional Distribution Function Using Bernstein Polynomial. Mathematics and Statistics, 12(2), 142 - 156. DOI: 10.13189/ms.2024.120204.