## Variance-Covariance Matrix Assumed Known

Consider the case of К = 2. We can write 0 = (0b 02)’ and 0O = (0ю, 02o)’ • It is intuitively reasonable that an optimal critical region should be outside some enclosure containing 0O, as depicted in Figure 9.9. What should be the specific shape of the enclosure?

An obvious first choice would be a circle with 0O at its center. That would amount to the test:

Reject HQ if (01 — 0ioC + (02 — 02oT > c

for some c, where c is chosen so as to make the probability of Type I error equal to a given value a. An undesirable feature of this choice can be demonstrated as follows: Suppose F0i is much larger than V02...

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