New classes of density estimates of low bias

Abstract

A desirable property of a kernel used for a density estimate is that it has high order as this gives low bias. Given a kernel K, a basis pj(z),j≥0, and q≥1, we show how to choose a=a0,…,aq−1′ such that K̂q(z)=K(z)∑j=0q−1ajpj(z) is a kernel of order at least q. a is given in terms of the mixed moments, EZjpk(Z). Some of the kernels involve the complex unit. A simulation study is performed to compare density estimates based on some of the proposed kernels to competitive ones including those in Withers and Nadarajah [1]. The proposed estimates are shown to have smaller biases and smaller mean squared errors.