Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

Abstract

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299-309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497-516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.

Adaptive Bayesian bandwidth selection in asymmetric kernel density estimation for nonnegative heavy-tailed data

“…These previous studies show that this approach is a very good alternative to classical methods. The present work is an extension of studies conducted by Brewer [3] and Zougab et al [23].…”

“…The posterior distribution and the Bayes estimators for each variable bandwidth can be derived exactly using the Bayes theorem and both quadratic and entropy loss functions. The Baysian approach with global, local and adaptive versions to bandwidth selection has received considerable attention, see [2,3,8] for univariate symmetric kernel density estimation, [12,13] for univariate kernel density estimation with censored data, [22,23] for discrete associated kernel estimators and [6,7,9,21,24] for multivariate kernel density estimation. These previous studies show that this approach is a very good alternative to classical methods.…”

“…The global kernel estimator tends to simultaneously under and over smooth f. This paper aims to (i) improve the global asymmetric kernel estimator (1) by considering the adaptive asymmetric kernel estimator using the BS-PE kernel in the estimation of pdfs of nonnegative HT data. The adaptive kernel estimator has been employed for example in [3] with Gaussian kernel and in [23] with associated binomial kernel. To our knowledge, this version has not been explored using asymmetric kernels, in particular for analyzing the nonnegative HT data.…”

Bayesian bandwidth selection in discrete multivariate associated kernel estimators for probability mass functions

“…Gangopadhyay and Cheung (2002), Kulasekera and Padgett (2006) and Kuruwita, Kulasekera, and Padgett (2010) proposed a Bayesian local bandwidth selection. For asymmetric kernels, we can consult Zougab et al (2012Zougab et al ( , 2013a; Zougab, Adjabi, and Kokonendji (2013b) who used discrete associated kernels. In the multivariate context, Zhang, King, and Hyndman (2006) presented MCMC method for the global bandwidth matrix selection using the symmetric Gaussian kernel; see also Zhang, King, and Shang (2013) for bandwidth selection in nonparametric regression model with mixed types of regressors.…”

Discrete triangular associated kernel and bandwidth choices in semiparametric estimation for count data

“…Note that for choosing bandwidth parameters in continuous nonparametric kernel estimation, one can refer to [5][6][7]. In addition, let us also remark that a Bayesian local approach is developed by Zougab et al [8,9] for bandwidth selection in discrete nonparametric associated kernel estimation of p.m.f. Finally, concerning count data, the problem of their semiparametric regression is treated by Abdous et al [10].…”