Multiple combined gamma kernel estimations for nonnegative data with Bayesian adaptive bandwidths

TL;DR

This paper introduces multivariate combined gamma kernel estimators with Bayesian adaptive bandwidths for nonnegative data, demonstrating their advantages over standard and modified kernels through theoretical analysis and real data applications.

Contribution

It proposes a novel multivariate combined gamma kernel approach with Bayesian adaptive bandwidth estimation, enhancing nonparametric smoothing of nonnegative data.

Findings

Outperforms pure standard and modified gamma kernels in simulations.

Provides asymptotic properties of the estimators.

Shows advantages in real data applications under specific criteria.

Abstract

A modified gamma kernel should not be automatically preferred to the standard gamma kernel, especially for univariate convex densities with a pole at the origin. In the multivariate case, multiple combined gamma kernels, defined as a product of univariate standard and modified ones, are here introduced for nonparametric and semiparametric smoothing of unknown orthant densities with support $[0,\infty)^d$. Asymptotical properties of these multivariate associated kernel estimators are established. Bayesian estimation of adaptive bandwidth vectors using multiple pure combined gamma smoothers, and in semiparametric setup, are exactly derived under the usual quadratic function. The simulation results and four illustrations on real datasets reveal very interesting advantages of the proposed combined approach for nonparametric smoothing, compare to both pure standard and pure modified gamma kernel versions, and under integrated squared error and average log-likelihood criteria.