Page 300 - Elements of Distribution Theory

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

286 Approximation of Integrals

Example 9.13. Consider approximation of the integral
∞

√ √
Q n (z) ≡ exp(x) nφ( nx) dx, z ∈ R,
z
for large n,as discussed in Example 9.12. We may use Theorem 9.16, taking h(x) = exp(x).
Then
∞

√ √ √ 1 exp(z) − 1 √
exp(x) nφ( nx) dx = [1 − ( nz)] 1 + O + √ φ( nz)
z n nz
and an approximation to Q n (z)isgiven by
√ exp(z) − 1 √
¯
Q n (z) = 1 − ( nz) + √ φ( nz).
nz
√
Suppose z = z 0 / n. Then
√
exp(z 0 / n) − 1 1 1
¯
Q n (z) = 1 − (z 0 ) + φ(z 0 ) = 1 − (z 0 ) + φ(z 0 )√ + O .
z 0 n n
√
Using the expansion for Q n (z)given in Example 9.12, it follows that, for z = z 0 / n,
¯
Q n (z) = Q n (z) + O(1/n)as n →∞,asexpected from Theorem 9.16.
Example 9.14 (Probability that a gamma random variable exceeds its mean). Let X
denote a random variable with a standard gamma distribution with index z. Then
1 ∞ z−1
Pr(X ≥ cE(X)) = t exp(−t) dt
(z) cz
where c is a positive constant. We consider approximation of the integral in this expression
for large values of z.
Using the change-of-variable y = t/z,we may write

∞ ∞
t z−1 exp(−t) dt = z z y z−1 exp(−zy) dy
cz c

∞ 1
= z z exp{−z(y − log(y))} dy.
c y
Hence, consider approximation of the integral
∞

1
exp{−z(y − log(y))} dy.
y
c
The first step in applying Theorem 9.16 is to write the integral in the form (9.4). Note
that the function y − log(y)is decreasing for y < 1 and increasing for y > 1 with minimum
value 1 at y = 1. Hence, consider the transformation
1
x = sgn(y − 1){2[y − log(y) − 1]} 2 .
This is a one-to-one function of y with
1 2
x = y − log(y) − 1
2
and
y
dy = xdx.
y − 1