Statistics and Its Interface

Volume 12 (2019)

Number 3

Recursive density estimators based on Robbins–Monro’s scheme and using Bernstein polynomials

Pages: 439 – 455



Yousri Slaoui (Université de Poitiers, Laboratoire de mathématiques et applications, Futuroscope Chasseneuil, France)

Asma Jmaei (Faculté des sciences de Bizerte, Université de Carthage, Jarzouna, Tunisia)


In this paper, we consider the alleviation of the boundary problem when the probability density function has bounded support. We apply Robbins–Monro’s algorithm and Bernstein polynomials to construct a recursive density estimator. We study the asymptotic properties of the proposed recursive estimator. We then compare our proposed recursive estimator with many others estimators. Finally, we confirm our theoretical result through a simulation study and then using two real datasets.


density estimation, stochastic approximation algorithm, Bernstein polynomial, smoothing, curve fitting

2010 Mathematics Subject Classification

Primary 62G07, 62L20. Secondary 65D10.

This work benefited from the financial support of the GDR 3477 GeoSto.

Received 15 July 2018

Published 4 June 2019