Gamma-weibull kernel estimation of the heavy tailed densities

TL;DR

This paper introduces a new gamma-weibull kernel estimator for heavy tailed densities on [0,∞), combining properties of gamma and weibull kernels to improve boundary and tail estimation with proven asymptotic properties.

Contribution

It proposes a novel gamma-weibull kernel estimator that effectively estimates heavy tailed densities, with theoretical analysis of its bias, variance, and optimal bandwidth.

Findings

The estimator has desirable boundary properties near zero.

The asymptotic bias and variance are derived.

Optimal bandwidth minimizes the mean integrated squared error.

Abstract

We consider the nonparametric estimation of the univariate heavy tailed probability density function (pdf) with a support on $[0,\infty)$ by independent data. To this end we construct the new kernel estimator as a combination of the asymmetric gamma and weibull kernels, ss. gamma-weibull kernel. The gamma kernel is nonnegative, changes the shape depending on the position on the semi-axis and possess good boundary properties for a wide class of densities. Thus, we use it to estimate the pdf near the zero boundary. The weibull kernel is based on the weibull distribution which can be heavy tailed and hence we use it to estimate the tail of the unknown pdf. The theoretical asymptotic properties of the proposed density estimator like bias and variance are derived. We obtain the optimal bandwidth selection for the estimate as a minimum of the mean integrated squared error (MISE). Optimal rate of convergence of the MISE for the density is found.