Bayesian Approach in Nonparametric Count Regression with Binomial Kernel

Recently, Kokonendji et al. have adapted the well-known Nadaraya–Watson kernel estimator for estimating the count function m in the context of nonparametric discrete regression. The authors have also investigated the bandwidth selection using the cross-validation method. In this article, we propose a Bayesian approach in the context of nonparametric count regression for estimating the bandwidth and the variance of the model error, which has not been estimated in Kokonendji et al. The model error is considered as Gaussian with mean of zero and a variance of σ². The Bayes estimates cannot be obtained in closed form and then, we use the well-known Markov chain Monte Carlo (MCMC) technique to compute the Bayes estimates under the squared errors loss function. The performance of this proposed approach and the cross-validation method are compared through simulation and real count data.     

A Note on Asymptotic Behavior of the Nonparametric Density Estimators in Multivariate Mixtures

The asymptotic behavior of the nonparametric density estimator has been given for a multivariate mixture model. It has been observed that the estimator is asymptotically normally distributed with bias of size h ² and variance of size (nh)^?1.     

Nonparametric Estimation of Variance Function for Functional Data Under Mixing Conditions

This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulation studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses.     

Central Limit Theorem for ISE of Kernel Density Estimators in Censored Dependent Model

In some long-term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the failure times and its kernel estimate f _n is the integrated square error(ISE). In this article, we derive a central limit theorem for the integrated square error of the kernel density estimators under a censored dependent model.     

Estimating additive models with missing responses

ABSTRACT

For multivariate regressors, the Nadaraya–Watson regression estimator suffers from the well-known curse of dimensionality. Additive models overcome this drawback. To estimate the additive components, it is usually assumed that we observe all the data. However, in many applied statistical analysis missing data occur. In this paper, we study the effect of missing responses on the additive components estimation. The estimators are based on marginal integration adapted to the missing situation. The proposed estimators turn out to be consistent under mild assumptions. A simulation study allows to compare the behavior of our procedures, under different scenarios.     

An optimization interpretation of integration and back-fitting estimators for separable nonparametric models

We provide an optimization interpretation of both back-fitting and integration estimators for additive nonparametric regression. We find that the integration estimator is a projection with respect to a product measure. We also provide further understanding of the back-fitting method.