The Estimation of Probability Densities and Cumulatives by Fourier Series Methods

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Abstract
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 coefficients of the density f. These M.I.S.E. expressions both facilitate the evaluation of error for specific densities and specific truncation points m and make it possible for an optimal value of m to be estimated from the sample. The simplicity of the M.I.S.E. expressions in the trigonometric case does not seem to be paralleled by simple expressions for Var âk, and hence M.I.S.E., for other orthogonal systems, notably those based upon Hermite polynomials. A procedure for choosing the optimal number of terms of m and m is derived and the M.I.S.E. of m is compared to that of several alternative estimators of the population density. It is shown that for almost all cumulatives there exists a finite m such that m has smaller M.I.S.E. than the sample cumulative, i.e., the step function.

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