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9. Confidence Interval Estimation 463

9.4 Multiple Comparisons

We first briefly describe some confidence region problems for the mean vec-
tor of a p–dimensional multivariate normal distribution when the p.d. disper-
2
sion matrix (i) is known and (ii) is of the form σ H with a known p × p matrix
H but σ is unknown. Next, we compare the mean of a control with the means
of independent treatments followed by the analogous comparisons among the
variances. Here, one encounters important applications of the multivariate
normal, t and F distributions which were introduced earlier in Section 4.6.

9.4.1 Estimating a Multivariate Normal Mean Vector

Example 9.4.1 Suppose that X , ..., X are iid p–dimensional multivariate
n
1
normal, N (µ, Σ), random variables. Let us assume that the dispersion matrix
p
Σ is p.d. and known. With given α ∈ (0, 1), we wish to derive a (1 – α)
confidence region for the mean vector µ. Towards this end, from the Theo-
rem 4.6.1, part (ii), let us recall that

Now, consider the non–negative expression as the
pivot and denote

We write for the upper 100α% point of the distribution, that is
,α
=1-α. Thus, we define a p–dimensional confidence region Q
for µ as follows:

One can immediately claim that

and hence, Q is a (1 – α) confidence region for the mean vector µ. Geometri-
cally, the confidence region Q will be a p–dimensional ellipsoid having its
center at the point , the sample mean vector. !