Light- and Heavy-Tailed Density Estimation by Gamma-Weibull Kernel

In our previous papers we focus on the gamma kernel estimators of density and its derivatives on positive semi-axis by dependent data by univariate and multivariate samples. We introduce the gamma product kernel estimators for the multivariate joint probability density function (pdf) with the nonnegative support and its partial derivatives by the multivariate dependent data with a strong mixing. The asymptotic behavior of the estimates and the optimal bandwidths in the sense of minimal mean integrated squared error (MISE) are obtained. However, it is impossible to fit accurately the tail of the heavy-tailed density by pure gamma kernel. Therefore, we construct the new kernel estimator as a combination of the asymmetric gamma and Weibull kernels, i.e. Gamma-Weibull kernel. The gamma kernel is nonnegative and it changes the shape depending on the position on the semi-axis and possesses 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 the bias and the variance are derived. We obtain the optimal bandwidth selection for the estimate as a minimum of the MISE. The optimal rate of convergence of the MISE for the density is found.