Abstract: Simulation studies provide four moment approximating
distributions to each of the four parameters of a beta distribution (Pearson
Type I). Two of the parameters refer to origin and scale, two to shape (skewness
and kurtosis). Type I random number generator is checked out, and the stability
of moments of random samples of size n over cycles; particular attention is paid to shape parameter
moments. In Type I region of validity (referred to skewness and kurtosis),
moment methods become unstable in the neighborhood of Type III line, and ultimately abort. Thus
extremely large variances and large higher moments arise.We probe the cause of this phenomenon. Simulation studies are turned to
since alternative power series methods are forbiddingly complicated. However,
use is made of the delta method to provide asymptotic variances of the
estimators, and asymptotic variances of percentage points of the basic
distribution. An account of work on the subject byK. Pearson, some of it a century ago, is given. In particular an
important paper by Pearson and Filon provides some estimates of probable errors
of moment parameter estimators such as the basic distribution parameters, the
mode, the skewness and others. The heated controversy between Pearson and Fisher
is considered.
Keywords and phrases: beta density, delta method, kurtosis variance, moments, Monte Carlo simulation, percentage points, skewness variance, variance of percentage points.
