Bayesian estimation of bandwidth in semiparametric kernel estimation of unknown probability mass and regression functions of count data

Abstract

This work takes advantage of semiparametric modelling which improves significantly in many situations the estimation accuracy of the purely nonparametric approach. Herein for semiparametric estimations of probability mass function (pmf) of count data, and an unknown count regression function (crf), the kernel used is a binomial one and the bandiwdth selection is investigated by developing Bayesian approaches. About the latter, Bayes local and global bandwidth approaches are used to establish data-driven selection procedures in semiparametric framework. From conjugate beta prior distributions of the smoothing parameter and under the squared errors loss function, Bayes estimate for pmf is obtained in closed form. This is not available for the crf which is computed by the Markov Chain Monte Carlo technique. Simulation studies demonstrate that both proposed methods perform better than the classical cross-validation procedures, in particular the smoothing quality and execution times are optimized. All applications are made on real data sets. B Nabil Zougab

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

“…However, in this paper and following Senga Kiessé et al (2015) and, Zougab et al (2014) we choose α = 0.5 and β = √ n. The choice of β which depends on the sample size n is important for the consistency of nonparametric kernel estimator of pmf and permits the convergence of Bayesian bandwidths estimators to zero (h j → 0 while n → ∞ for j = 1, . .…”

“…Therefore, recently the so-called discrete associated kernel method is largely developed by several authors; see, e.g., Kokonendji and Senga Kiessé (2011), Kokonendji, Senga Kiessé, and Balakrishnan (2009), Kokonendji et al (2007), Wansouwé, Kokonendji, and Kolyang (2015), Zougab, Adjabi, and Kokonendji (2012) and Zougab, Adjabi, and Kokonendji (2013a). Note that the semi-parametric estimation of pmf and count rf (crf) is also investigated, see for example Abdous, Kokonendji, and Senga Kiessé (2012) and Senga Kiessé, Zougab, and Kokonendji (2015). They showed that the use of a discrete associated kernel is more appropriate than the use of a continuous kernel for both estimations of pmf and crf of a discrete variable.…”

Ake: An R Package for Discrete and Continuous Associated Kernel Estimations

“…However, the naive estimator is appropriate only when the sample size is large, and the discrete kernel of Aitchison and Aitken (1976) is only suitable for categorical data; see Hayfield and Racine (2008) and also Hayfield and Racine (2014). Kokonendji et al (2009) adapted the Nadaraya (1964) and Watson (1964) kernel to the discrete unknown function m, using the discrete associated kernels. In their work, using the integrated mean square error and the coefficient of determination R 2 , they showed that the binomial or discrete triangular kernels are better compared to the optimal Epanechnikov kernel.…”

“…where m −i (X i ) is computed as m n of (24) excluding X i ; see, e.g., Kokonendji et al (2009). The hcvreg.fun function to compute this optimal bandwidth is described in Table 3.…”

On finite sample properties of nonparametric discrete asymmetric kernel estimators

“…f of 1 i.i.d. observations (X i ) i=1,••• ,n was constructed to behave asymptotically as the frequency estimator F(x) = n −1 n i=1 1 {x} (X i ), x ∈ S, where 1 A denotes the indicator function of the set A (for details about f , see later equation (6) in Section 3).…”