Approximations for skewed probability densities based on Laguerre series and biological applications

Abstract

The gamma and chi-squared distributions have been widely used in statistics, both as a fit to experimental data, particularly lifetime data, and as limiting distributions of a test statistic. The special case of an exponential distribution also arises as some parameter approaches a limit, for example, when the threshold approaches infinity in a first passage time problem. Here we seek to efficiently approximate the behaviour of distributions that are close to a gamma distribution. The approximation is of the Gram-Charlier type with the parent distribution being a gamma distribution, rather than a normal distribution, and with Laguere polynomial multiplers rather Hermite polynomial multiplers. The Laguerre series approximation (L-series) is then developed and several application areas are considered. The problem of estimation of the coefficients in the approximation is then considered by several methods: method of momemnts, method of L-moments, and maximun likelihood estimation. Finally a graphical diagnostic procedure to examine when applying L-series might be appropriate. (This talk is based on joint work with Te Hsin LUNG.)