TESTING ABOUT A VECTOR PARAMETER
Those who are not familiar with matrix analysis should study Chapter 11 before reading this section. The results of this chapter will not be needed to understand Chapter 10. Insofar as possible, we shall illustrate our results in the two-dimensional case.
We consider the problem of testing H0: 0 = 00 against Ну 0 Ф 0O, where 0 is a A-dimensional vector of parameters. We are to use the test statistic 0 ~ N(Q, X), where X is а К X К variance-covariance matrix: that is, X = £(0 — 0)(0 — 0)’. (Throughout this section a matrix is denoted by a boldface capital letter and a vector by a boldface lower-case letter.) In
FIGURE 9.9 Critical region for testing about two parameters
Section 9.7.1 we consider the case where X is completely known, and in Section 9.7.2 the case where X is known only up to a scalar multiple.