Nonparametric density estimation for multivariate bounded data using two non-negative multiplicative bias correction methods

Abstract

In this article we propose two new Multiplicative Bias Correction (MBC) techniques for nonparametric multivariate density estimation. We deal with positively supported data but our results can easily be extended to the case of mixtures of bounded and unbounded supports. Both methods improve the optimal rate of convergence of the mean squared error up to O(n −8/(8+d) ), where d is the dimension of the underlying data set. Moreover, they overcome the boundary effect near the origin and their values are always non-negative. We investigate asymptotic properties like bias and variance as well as the performance of our estimators in Monte Carlo Simulations and in a real data example.

A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications

“…In [8-12, 16, 19, 29, 58], many asymptotic properties for Gamma kernel estimators of density functions supported on the half-line [0, ∞) were studied, among other things: pointwise bias, pointwise variance, mean squared error, mean integrated squared error, asymptotic normality and uniform strong consistency. Also, bias reduction techniques were explored by Igarashi and Kakizawa [31] and Funke and Kawka [21], and adaptative Bayesian methods of bandwidth selection were presented by Somé [52] and Somé and Kokonendji [53].…”

Bayesian Bandwidths in Semiparametric Modelling for Nonnegative Orthant Data with Diagnostics

“…Once again, the bias is different from that of (10). Thus, the proposed semiparametric estimator f n in (16) of f can be shown to be better (or not) than the traditional nonparametric one f n in (8). The following subsection provides a practical solution.…”

“…The nonparametric topic of associated kernels, which is adaptable to any support T + d of probability density or mass function (pdmf), is widely studied in very recent years. We can refer to [6][7][8][9][10][11][12][13][14][15] for general results and more specific developments on associated kernel orthant distributions using classical cross-validation and Bayesian methods to select bandwidth matrices. Thus, a natural question of flexible semiparametric modeling now arises for all these multivariate orthant datasets.…”

Multiplicative bias correction for asymmetric kernel density estimators revisited

“…for the TS a -type bias-corrected beta KDE (the TS 1 -type or JF a -type bias-corrected beta KDE was additionally studied by Igarashi (2016a)). However [3] , as will be revisited in this paper, it turns out that Hirukawa's asymptotic variance miss two terms, and, to make matters worse, Hirukawa's original incorrect proof may lead to similar incorrect conclusions in his companion papers (Hirukawa andSakudo (2014, 2015)) and subsequent papers (Funke and Kawka (2015), Zougab and Adjabi (2016), and Zougab et al (2018)). To be exact, as Jones et al (1995) did for the standard KDE (S = R), we need, for the variance manipulation, to look at four terms from the law of total variance/covariance, in which only one term is negligible, while other three terms contribute to the final result (i.e., the aforementioned authors's asymptotic variances would be incorrectly asserted in common).…”

“…(the formula given by Funke and Kawka (2015) (see also Hirukawa (2010)) miss the factor λ d ). Using 2/(d+8) , which is feasible if d < 8 (see Assumption (iii d )), the convergence rate n −8/(d+8) is achieved.…”