Asymptotically Pearson's Transformations

The paper deals with a refinement of some limit theorems and asymptotic formulas similar to the Edgeworth and Cornish–Fisher expansion. In the first part of the paper various aspects of approximations for distributions close to those of Pearson’s family are discussed. The theory of asymptotically Pearson transformations of random variables generalizes the theory of asymptotically normal transformation [1], [2]. These results are used in part two to study asymptotic properties of the B-distribution that enable one to construct a new approximation for the binomial distribution which is more exact than the normal or the Poisson approximation. The third part of the paper deals with examples of asymptotically Pearson transformations. Here, in particular, approximation problems for non-central distributions $\chi ^2 $, $F$ and $t$, as well as the Kolmogorov–Smirnov test distribution are discussed.