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

Abstract

a b s t r a c tThis paper proposed a nonparametric estimator for probability mass function of multivariate data. The estimator is based on discrete multivariate associated kernel without correlation structure. For the choice of the bandwidth diagonal matrix, we presented the Bayes global method against the likelihood cross-validation one, and we used the Bayesian Markov chain Monte Carlo (MCMC) method for deriving the global optimal bandwidth. We have compared the proposed method with the cross-validation method. The performance of both methods is evaluated under the integrated square error criterion through simulation studies based on for univariate and multivariate models. We also presented applications of the proposed methods to bivariate and trivariate real data. The obtained results show that the Bayes global method performs better than cross-validation one, even for the Poisson kernel which is the very bad discrete associated kernel among binomial, discrete triangular and Dirac discrete uniform kernels.

Smoothed Estimators for Markov Chains with Sparse Spatial Observations

“…Therefore, the dependence structure between d categorical variables needs to be incorporated in the smoothing process. It could be achieved either by extending the current productive multivariate discrete kernel smoothers or by expanding the d ‐dimensional smoothing vector to a ( d , d ) smoothing matrix whose nondiagonal elements control the form of orientation of the kernel (Belaid et al 2016). For the classic Markov chain model, the dependence structure refers to the temporal dependence between category at t −1 and category at t , while for the spatial Markov chain model, the temporal dependence, the cross‐sectional dependence (category at t −1 and spatial lag category at t −1) as well as the spatiotemporal dependence (spatial lag category at t −1 and category at t ) need to be incorporated.…”

“…Selecting an appropriate smoothing parameter is of paramount importance to the performance of the discrete kernel estimators. Approaches could be similar to those for continuous kernel estimation which has received more attention, including the plug‐in method (Chu, Henderson, and Parmeter 2015), the cross‐validation method (Henderson and Parmeter 2015), and the Bayesian method (Agresti and Hitchcock 2005; Belaid et al 2016). The plug‐in bandwidth for a one‐way contingency table is derived by minimizing mean squared error (MSE) summed over the sample space.…”

“…Ouyang, Li, and Racine (2006) demonstrate that the discrete kernel estimators with the cross‐validated smoothing parameters perform better than MLEs in terms of the summed MSE based on some Monte Carlo simulations. Belaid et al (2016) propose the Bayesian Markov chain Monte Carlo (MCMC) method through the likelihood cross‐validation criterion for selecting the optimal smoothing parameters and compare it with LSCV. They find a better performance of the Bayesian MCMC method, but obviously, it is much more computationally intensive.…”

On finite sample properties of nonparametric discrete asymmetric kernel estimators

“…f . Based on the ranking of variance of discrete standard kernels, equation (8) shows that using binomial kernel provides smaller estimator bias than Poisson and negative binomial kernels. The variance term can be majored as follows:…”