Approximating Probability Distributions Using Moments - Davis, Charles Shaw (My University, 1977)
We study the problem of finding approximate significance points of random variables whose exact distributions are unknown or extremely complicated . We consider the case where at least the first three moments, and possibly the lower or upper endpoint of the distribution are known.
The methods of approximation studied include the Johnson system of transformations, Pearson curves, Pearson curves with known lower terminal, Cornish-Fisher expansions and the approximation a+bW, where W is chi-squared with p degrees of freedom . A new three-moment approximation of the form (cW)^k, with W as defined above, is also considered. These methods of approximation are discussed, with special attention to fitting procedures and computer implementation.
The methods of approximation are compared, with respect to ease of application and accuracy of approximation, over a wide variety of exact distributions. The accuracy of each approximation is discussed and guidelines are given for determining which of several approximations should be used in a particular case.
Keywords: approximate significance points of random variables, random variables, approximatae significance, distributions, Pearson curves, Cornish-Fisher, a+bW, chi-squared, degrees of freedom, methods of approximation