Page 271 - Elements of Distribution Theory

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9

Approximation of Integrals

9.1 Introduction
Integrals play a fundamental role in distribution theory and, when exact calculation of an
integral is either difficult or impossible, it is often useful to use an approximation. In this
chapter, several methods of approximating integrals are considered. The goal here is not
the determination of the numerical value of a given integral; instead, we are concerned with
determining the properties of the integrals that commonly arise in distribution theory. These
properties are useful for understanding the properties of the statistical procedures that are
based on those integrals.

9.2 Some Useful Functions

There are a number of important functions that repeatedly appear in statistical calculations,
such as the gamma function, the incomplete gamma function, and the standard normal distri-
bution function. These functions are well-studied and their properties are well-understood;
when an integral under consideration can be expressed in terms of one of these functions,
the properties of the integral are, to a large extent, also well-understood. In this section,
we consider the basic properties of these functions; further properties are presented in the
remaining sections of this chapter.

Gamma function
The gamma function is defined by

∞

(x) = t x−1 exp(−t) dt, x > 0.
0
The most important property of the gamma function is its recursion property:
(x + 1) = x (x), x > 0.

This, together with the fact that (1) = 1, shows that

(n + 1) = n!, n = 0, 1, 2,...
so that the gamma function represents a generalization of the factorial function to non-
integer positive arguments. These properties are formally stated in the following theorem.

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