On Multivariate Density Estimates Based on Orthogonal Expansions

Abstract

The topics of orthogonality and Fourier series occupy a central position in analysis. Nevertheless, there is surprisingly little statistical literature, with the exception of that of time series and regression, which involves Fourier analysis. In the last decade however, several papers have appeared which deal with the estimation of orthogonal expansions of distribution densities and cumulatives. Cencov [1] and Van Ryzin [6] considered general properties of orthogonal expansion based density estimators and the latter applied these properties to obtain classification procedures. Schwartz [3] and the authors [2] and [4] investigated respectively the Hermite and Trigonometric special cases. The authors also obtained certain general results which apply not only to estimators of the population density but also to estimators of the population cumulative [2], [4] and [5]. In this paper several results derived for the univariate case are extended to the multivariate case. Also a new relationship is obtained which involves general Fourier expansions and estimators. Although there is some reason for calling the Gram-Charlier estimation of distribution densities a Fourier method, one fundamental aspect of Fourier methods is not shared by Gram-Charlier estimation. Gram-Charlier techniques make no use of Parseval's Formula or related error relationships of Fourier analysis. The ease with which the mean integrated square error (MISE) is evaluated, when Fourier methods are applied, accounts for most of the recent interest in this area. Section 1 of this paper deals with an investigation of two general MISE relationships for multivariate estimates of Fourier expansions. The relationship given in Theorem 2 is particularly simple and yet includes the four MISE's which are involved in the estimation problem. In Section 2 the choice of orthogonal functions is restricted to the trigonometric polynomials. It is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated. This result is of considerable utility since it yields a rule for deciding which terms should be included in the estimate of the multivariate density.