On the Roy-Tiku Approximation to the Distribution of Sample Variances from Nonnormal Universes

Abstract

Some justifications for the Roy-Tiku (1962) approximation to the distributions of sample variances by Laguerre polynomial series are provided. It is shown that if one approximates any nonnegative continuous random variable by a finite series of Laguerre polynomials up to the kth degree, then the first k moments of the approximated distribution are equal to the first k moments of the random variable. We also derive an alternative Laguerre series approximation, including the Box (1953) approximation as a special case.

Key Words:

• Box approximation
• Mixtures of normals
• Moments
• Departure from normality
• Laguerre polynomial
• Roy-Tiku approximation
• Sample variance