Polynomial Multivariate Density Estimates and Approximants

It is often the case that the exact moments of a continuous distribution whose support is confined to a closed interval can be explicitly determined, while its exact density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. A representation of a least-squares approximating polynomial which does not rely on orthogonal polynomials, is obtained in this paper. The resulting density approximant readily lends itself to numerical evaluation and algebraic manipulations. In an application, the distribution of the square of the distance between two random points in the unit cube is approximated—for comparison purposes a closed form representation of the exact density function is derived. The proposed approach can also be employed in the context of density estimation; in this case, sample moments are being used along with an extended range taken to be the support of a preliminary estimate which is subsequently truncated and normalized. This density estimation methodology is applied to the ‘Buffalo snowfall data’. Alternatively, one can use such density approximants for smoothing initial density estimates. To illustrate this methodology, we introduce a new type of histograms which, unlike averaged shifted histograms, are adaptive. Extensions to multivariate distributions are discussed and applications involving lagged observations from the Old Faithful eruption durations data set as well as a trivariate Dirichlet distribution are presented.