On improving convergence rate of Bernstein polynomial density estimator

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

“…and then the corresponding M ISE M ISE f n,m,m/2 = 9 32C 8 1 C 2 C 8 3 1/9 8 n −8/9 + o n −8/9 . Igarashi and Kakizawa (2014) have generalized the estimator proposed by Leblanc f n,m,m/2 and defined in (17)…”

“…This estimator was considered by Igarashi and Kakizawa (2014) by applying a multiplicative bias correction method to the Bernstein estimator…”

“…For a smooth estimate of a density function with finite known support, Vitale's method (Vitale (1975)) based on the Bernstein polynomials, illustrated below, also has been investigated in the literature (Ghosal (2001), Babu et al (2002), Kakizawa (2004), Rao (2005)) and more recently by Leblanc (2010) and Igarashi and Kakizawa (2014)). The idea comes from the Weierstrass's approximation theorem that for any continuous function u on the interval [0, 1], we have…”

Design‐based inference on Bernstein type estimators for continuous populations

“…Guan, 2016;Igarashi & Kakizawa, 2014;Leblanc, 2012;Manté, 2015;Turnbull & Ghosh, 2014 and references therein). Guan, 2016;Igarashi & Kakizawa, 2014;Leblanc, 2012;Manté, 2015;Turnbull & Ghosh, 2014 and references therein).…”

“…In the last decades the use of the Bernstein polynomial has become more frequent in statistics, especially for estimating distribution and density functions (see e.g. Guan, 2016;Igarashi & Kakizawa, 2014;Leblanc, 2012;Manté, 2015;Turnbull & Ghosh, 2014 and references therein). However, to our knowledge, this paper is the first attempt to use Bernstein polynomials for estimating the values of the function by recording the function values on the sampled points instead of on purposively selected points, thus allowing not only the surface estimation but also the evaluation of the precision of the resulting estimates.…”