The approximate distribution of four moment statistics from type III distributions

Taylor series in the sample size are set up for the first four moments of the standard deviation, skewness, kurtosis, and coefficient of variation, the populations being x 2 (gamma, Pearson Type III). These moments being out of reach of purely mathematical development, the study proceeds along two independent lines. For the one, simulation methods are used, an attempt being made to fix a cycle length to ensure some stability -this cycle length is pivoted on the fourth moment of the kurtosis, an expression involving sixteenth powers of the basic x 2 - random variable. The second line of attack uses the Taylor moment series (which are taken out to at most sixty terms in the total derivatives). An algorithm is used to derive the expectation of a product of powers of elements which consist of non-central sample deviates; there are four of these involved in the kurtosis, three in the skewness, and two in the standard deviation. There is an added parameter for sample size. This expectation of products of powers of sample deviates generates a set of coefficients, each coefficient multiplied by a power of n 1 the larger the moment product, the greater is the span of the powers of n 1 . If a final moment series is desired to include all contributions up to n s , then at least 2s terms will be required in the Taylor expansion; moreover the series turn out to be divergent, as far as can be judged by the behavior of the terms computed. At this point, since the series are not seen to be onesigned, and since divergence is not too chaotic (as fast as the triple factorial, say), rational fraction sequences are set up to dilute divergence (or accelerate apparent convergence); the approach is often successful but there are problems with small sample sizes and large skewness of the population sampled. Lastly, gross errors in relying on basic asymptotes are noted. The study brings out unusual confluences - computer oriented numerical analysis, distributional theory and approximation, and the power of rational fractions as divergency reducing tools.