Page 293 - Elements of Distribution Theory

P1: JZP
052184472Xc09 CUNY148/Severini June 2, 2005 12:8

9.5 Laplace’s Method 279

Example 9.9 (Ratio of two integrals). Consider approximation of the ratio
b

a h 1 (y)exp{ng(y)} dy
b
a h 2 (y)exp{ng(y)} dy
where g and both h 1 and h 2 satisfy the conditions of Theorem 9.14, with |h 1 (ˆ y)| > 0 and
|h 2 (ˆ y)| > 0. Here ˆ y is the maximizer of g and must satisfy a < ˆ y < b.
Since, as n →∞,
b √
(2π)h j (ˆ y) −1

h j (y)exp{ng(y)} dy = exp{ng(ˆ y)} 1 [1 + O(n )], j = 1, 2,

a [−ng (ˆ y)] 2
it follows that
b

a h 1 (y)exp{ng(y)} dy h 1 (ˆ y) −1
b = h 2 (ˆ y) [1 + O(n )], as n →∞.
a h 2 (y)exp{ng(y)} dy
Example 9.10. Let Y denote a real-valued random variable with an absolutely continuous
distribution with density function

p(y; a) = c(a)exp{−2a cosh(y)}, −∞ < y < ∞;
here a is a nonnegative constant and c(a)isa constant chosen so that the density integrates to
1. This distribution arises in the following manner. Let X 1 and X 2 denote independent iden-
1
tically distributed gamma random variables, let Y = log(X 1 /X 2 )/2 and let A = (X 1 X 2 ) 2 .
Then the conditional distribution of Y,given A = a, has the density p(y; a).
Consider approximation of the constant c(a) for large values of a;todo this, we need to
approximate the integral

∞
exp{−2a cosh(y)} dy.
−∞

We may apply Theorem 9.14 with g(y) =−2 cosh(y) and h(y) = 1. Hence, ˆ y = 0, g (ˆ y) =
−2, and
√
∞
(2π) −1

exp{−2a cosh(y)} dy = exp(−2a) √ [1 + O(a )] as a →∞.
(2a)
−∞
It follows that
√
a −1
c(a) = √ exp(2a)[1 + O(a )] as a →∞.
π
Laplace’s method, as given in Theorem 9.14, applies to integrals of the form
b
h(y)exp{ng(y)} dy
a
in which the maximum of g occurs at an interior point of (a, b); then, after changing the
variable of integration, the integral can be approximated by Corollary 9.1. If the maximum
of g occurs at either a or b, the same general approach may be used; however, in this case,
the approximation is based on Theorem 9.12. The following result considers the case in
which the maximum occurs at the lower endpoint a;a similar result may be derived for the
case in which the maximum occurs at the upper endpoint.