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The Bayesian approach to bandwidth selection in discrete associated kernel estimation of probability mass function is a very good alternative to the classical popular methods such as the methods which adopt the asymptotic mean integrated squared error as a criterion and the cross-validation technique. In this paper, we propose a Bayesian local approach to bandwidth selection considering the binomial kernel estimator and locally treating the bandwidth h as a random quantity with a prior distribution. The local bandwidth is estimated by the posterior mean of h. The performance of this proposed approach and that of the classical methods are compared using simulations of data generated from known discrete functions. The new method is then applied to a real count data set. The smoothing quality of the Bayes estimator is very satisfactory.

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In this paper, we consider an interesting problem on adaptive Birnbaum-Saunders-power-exponential (BS-PE) kernel density estimation for nonnegative heavy-tailed (HT) data. Treating the variable bandwidths h i , i = 1, . . . , n of adaptive BS-PE kernel as parameters, we then propose a conjugate prior and estimate the h i 's by using the popular quadratic and entropy loss functions. Explicit formulas are obtained for the posterior and Bayes estimators. Comparison simulations with global unbiased cross-validation bandwidth selection technique were conducted under four HT distributions. Finally, two applications based on HT real data are presented and analyzed.