Chapter 36: Expansions of a class of cumulative distribution functions

In this chapter we show how to transform a selection of well-known distribution functions, such as the gamma and beta distributions, into a standard form. We derive asymptotic expansions with respect to one parameter, and the expansion is uniformly valid with respect to a second parameter. The standard form is a convenient starting point in several cases, however, we will see that for some examples it is better to use contour integral representations. This will be explained in Chapter 37 for the incomplete gamma functions, in Chapter 38 for the incomplete beta functions, and in Chapter 39 for the non-central chi-square distribution functions (or Marcum functions). In a final chapter we consider the problem of inverting the cumulative distribution functions by using asymptotic methods.