Local orthogonal polynomial expansions for density estimation
Dassanayake, Placida D.A.
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We propose a new method to estimate the density function of a univariate continuous random variable. local orthogonal polynomial expansions (LOrPE) the new method draws similarities with kernel density estimation (KDE), orthogonal series density estimation (OSDE) and local likelihood density estimation (LLDE). It is most similar to LLDE in that it is a local method where the approximation is obtained at each point of the support. It is connected to the OSDE in that it is constructed by using an orthogonal polynomial series expansion at each point of the support. The order of the series (M) used is one of the method's tuning parameters, a localized version of OSDE. Finally, LOrPE utilizes a bandwidth (h), the second tuning parameter, in order to construct the orthogonal polynomials over a localized window, and in this respect it is similar to KDE. Also, we show that under certain conditions, LORPE is equivalent to KDE with a high order kernel. Comparisons of LOrPE with KDE are performed under a variety of conditions. We find that in terms of MISE LOrPE performs better that KDE when estimating densities with sharp boundaries and both LOrPE and KDE results remain same when estimating densities which quickly decay to zero at infinity.