Page 488 - Probability and Statistical Inference

9. Confidence Interval Estimation 465

Let us denote

2
which respectively estimate µ and σ . Recall from Theorem 4.6.1, part (ii)
that

One can show easily that
2
(pn – p)S /s is distributed as and that the mean
2
vector and S are independently distributed. (9.4.6)
2
Combining (9.4.5) and (9.4.6) we define the pivot

But, we note that we can rewrite U as

so that U has the F p,pn–p distribution. We write F p,pn–p,α for the upper 100α%
point of the F p,pn–p distribution, that is P{U < F p,pn–p,α } = 1 – α. Thus, we
define a p–dimensional confidence region Q for µ as follows:

Thus, we can immediately claim that

and hence Q is a (1 – α) confidence region for the mean vector µ. Again,
geometrically the confidence region Q will be a p–dimensional ellipsoid with
its center at the point . !

9.4.2 Comparing the Means

Example 9.4.4 Suppose that X , ..., X are iid random samples from the N(µ ,
i
i1
in
2
σ ) population, i = 0, 1, ..., p. The observations X , ..., X refer to a control
01
0n
th
with its mean µ whereas X , ..., X are the observations from the i treat-
0
in
i1
ment, i = 1, ..., p. Let us also assume that all the observations from the treat-
ments and control are independent and that all the parameters are unknown.