It is often the case that the exact moments of a statistic of the continuous type can be explicitly determined, while its density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. The density approximants discussed in this article are based on the first
exact moments of the corresponding distributions. A unified semiparametric approach to density approximation is introduced. Then, it is shown that the resulting approximants are mathematically equivalent to those obtained by making use of certain orthogonal polynomials, such as the Legendre, Laguerre, Jacobi, and Hermite polynomials. Several examples illustrate the proposed methodology.
2. Approximants Based on Legendre Polynomials
3. Approximants Based on Laguerre Polynomials
4. A Unified Methodology
About the Author
Serge B. Provost is a professor of statistics in the Department of Statistical and Actuarial Sciences at The University of Western Ontario. He holds a Ph.D. in mathematics and statistics from McGill University. He is a fellow of the Royal Statistical Society and an associate of the Society of Actuaries. He co-authored two research monographs on quadratic and bilinear forms in random variables. His research interests include distribution theory, multivariate analysis, order statistics, time series, and computational statistics.
Serge B. Provost
Department of Statistical and Actuarial Sciences
The University of Western Ontario
London, Ontario, Canada N6A 5B7