Polynomial Multivariate Density Estimates and Approximants

• Activity Number: 71
• Type: Contributed
• Date/Time: Monday, August 12, 2002 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics*
• Author(s): Serge Provost*+
• Affiliation(s): University of Western Ontario
• Keywords: Density estimation ; Density approximation ; Multivariate distributions ; Nonparametric estimation ; Multidimensional data sets ; Multivariate histograms

Abstract

It is often the case that the moments of a univariate or multivariate distribution can be readily determined while its exact density function turns out to be mathematically intractable. It will be shown that the density function of a continuous distribution defined in a compact region can be easily approximated from its joint moments by solving a linear system involving a Hilbert-type matrix. When sample moments are being used, in which case an extension of the domain of the empirical distribution is indicated, the same systems of linear equations will also yield polynomial density estimates. Alternatively, the moments of some preliminary density estimate, such as an averaged shifted histogram with optimal bin width, initially could be used in lieu of the sample moments to produce smooth bona fide density estimates. The proposed methodology will be applied to the set of bivariate points of the lagged Old Faithful eruption duration data, and a polynomial density approximant will be obtained for a Dirichlet random vector.