A Numerical Study of Small Parameter Behavior of Some Families of Distributions

Limiting distributions play an important role in approximating the exact distributions, especially when they have a rather cumbersome analytic form, or simply when they do not have a closed from. The question that naturally arises is how good the approximation is. In this article, we propose a procedure for the numerical assessment of the “goodness” of some easy-to-calculate limiting distributions, originally proposed in Bar-Lev and Enis, in various cases of the underlying distributions, some of which are inherently computationally challenging. The details of the procedure are provided in three examples. The first example deals with the gamma distributions; the second deals with Bessel distributions related to a symmetric random walk, and the third example deals with positive stable distributions. The details of two additional variations of these examples are also discussed. These examples illustrate the ease with which the limiting approximations could be applied in the various cases, well-demonstrating their computational simplicity and attractiveness.