Page 582 - Probability and Statistical Inference

12. Large-Sample Inference 559

α = .05, p = .6, .7 and n = 30.
Table 12.4.1. Comparison of Ten Approximate 95% Confidence
Intervals Based on (12.3.18) and (12.4.4)

p = .6 and z = z = 1.96 p = .7 and z = z = 1.96
.025 .025
# a a b b a a b b
L U L U L U L U
1 .3893 .7440 .3889 .7360 .5751 .8916 .5637 .8734
2 .4980 .8354 .4910 .8212 .4609 .8058 .4561 .7937
3 .3211 .6789 .3249 .6751 .4980 .8354 .4910 .8212
4 .4980 .8354 .4910 .8212 .6153 .9180 .6016 .8980
5 .3893 .7440 .3889 .7360 .6153 .9180 .6016 .8980
6 .3211 .6789 .3249 .6751 .5751 .8916 .5637 .8734
7 .3548 .7119 .3565 .7060 .6569 .9431 .6409 .9211
8 .4247 .7753 .4221 .7653 .6153 .9180 .6016 .8980
9 .4609 .8058 .4561 .7937 .6153 .9180 .6016 .8980
10 .4609 .8058 .4561 .7937 .4980 .8354 .4910 .8212

Based on this exercise, it appears that the intervals (b , b ) obtained via
L
U
variance stabilizing transformation are on the whole a little “tighter” than the
intervals (a , a ). The reader may perform large-scale simulations in order to
U
L
compare two types of confidence intervals for a wide range of values of p
and n.
12.4.2 The Poisson Mean
Suppose that X , ..., X are iid Poisson(λ) random variables where 0 < λ <
n
1
∞ is the unknown parameter. The minimal sufficient estimator of λ is
denoted by One has Even
though as n → ∞, it will be hard to use
as a pivot o construct tests and confidence intervals for λ.
See the Exercise 12.3.14. Since the normalizing constant in the denominator
depends on the unknown parameter λ, the “power” calculations will be awk-
ward too.
We may invoke Mann-Wald Theorem from (12.4.1) and require a suitable
function g(.) such that the asymptotic variance of becomes
free from λ. In other words, we want to have

that is