Kernel Density Estimation For Heavy-Tailed Distributions Based On Transformations

This article mainly studies how to solve the density estimation problem of heavy-tailed dis-tribution in Rn space by transformation method.For one-dimensional fat-tailed distribution,we change the transformation function of the original quadratic transformation method and propose a new estimation methodology,Kernel Inverse Polynomial Modified Champernowne Estimator(KIPMCE).We prove that this estimator has some good asymptotic properties,and its effective-ness is verified by real data and stochastic simulation.For n-dimensional fat-tailed distribution,we extend the range of transformation from the Rn space to the n-dimensional Riemannian man-ifold.Firstly,we improve the asymptotic theory of kernel estimator on manifolds in the historical literature.Then,We have deduced in detail the relationship between two density functions,one of which is about the original sample and the other one is with regard to the transformed sam-ple.Based on this relationship and the estimated density of transformed sample,a new density estimator for multidimensional fat-tailed distribution in Rn space,called Kernel Stereographic Projection Transformation Density Estimator(KSPTDE),is proposed.