The Estimation of Probability Densities and Cumulatives by Fourier Series Methods

A class of estimators (referred to as the Fourier estimators m and m) of the probability density function f and the associated cumulative distribution function F are considered. Here m = Σmk=0 âkψk and m = Σmk=0 Âk ψk where the functions {ψk} comprise an orthogonal set with respect to weight function w(x), and the statistics âk and Âk are formed from the n unordered observations.Simple expressions are found for the mean integrated square errors, M.I.S.E., of the estimators m and m, i.e., E∫{ƒ(x) – m(x)}2ω(x)dx and E∫{F(x) – m(x)}2w(x)dx in terms of the variances of âk and Âk and the Fourier coefficients of f and F.For Fourier estimators based upon the trigonometric orthogonal functions the âk are the sample trigonometric moments. The variances and covariances of the statistics âk and Âk for these special cases are shown to be linear functions of the density f's Fourier coefficients. Therefore, simple expressions are obtained which relate the M.I.S.E. of the Fourier estimators m and m to the Fourier coeffi...

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