Moment-Based Density Approximants

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 n 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.