Page 491 - Probability and Statistical Inference

468 9. Confidence Interval Estimation

th
refer to the i treatment, i = 1, ..., p. Let us also assume that the all observa-
tions from the treatments and control are independent.
2
2
The problem is one of comparing the treatment variances σ ,...,σ with
p
1
the control variance σ by constructing a joint confidence region of the para-
2
0
metric function Let us denote
the i sample variance which estimates i = 0, 1, ..., p. These sample
th
variances are all independent and also is distributed as i =
0, 1, ..., p. Next, let us consider the pivot
which has a multivariate F distribution, M F (v , v , ..., v ) with v = v = ...
0
p
p
0
1
1
= v = n – 1, defined in Section 4.6.3. With σ = (σ , σ , ..., σ ), one may find
0
p
1
p
a positive number b = b p,n,α such that
The Tables from Finney (1941) and Armitage and Krishnaiah (1964) will pro-
vide the b values for different choices of n and α. Next, we may rephrase
(9.4.18) to make the following joint statements:

The simultaneous confidence intervals given by (9.4.19) jointly have 100(1 –
α)% confidence. !
Example 9.4.7 (Example 9.4.6 Continued) From the pivot and its distribu-
tion described by (9.4.17), it should be clear how one may proceed to obtain
the simultaneous (1 – α) two–sided confidence intervals for the variance
ratios i = 1, ..., p. We need to find two positive numbers a and b , a <
b, such that

so that with σ′ = (s , s , ..., s ) we have
p
0
1
Using (9.4.20), one can obviously determine simultaneous (1 – α) two–sided
confidence intervals for all the variance ratios i = 1, ..., p. We leave the
details out as Exercise 9.4.4.