ABSTRACT
This chapter deals with testing problems concerning mean vectors of multivariate distributions. Using the same developments of the appropriate test criteria we will also construct the confidence region for a mean vector. It will not be difficult for the reader to construct the confidence regions for the other cases discussed in this chapter. The matrix S is rarely known in most practical problems and tests of hypotheses concerning the mean vectors must be based on an appropriate estimate of S. However, in cases of long experience with the same experimental variables, we can sometimes assume S to be known. In deriving suitable test criteria for different testing problems we shall use mainly the well-known likelihood ratio principle and the approach of invariance as outlined in Chapter 3. The heuristic approach of Roy’s union-intersection principle of test construction also leads to suitable test criteria. We shall include it as an exercise. For further material on this the reader is referred to Giri (1965), books on multivariate analysis by Anderson (1984), Eaton (1988), Farrell (1985), Kariya (1985), Kariya and Sinha (1989), Muirhead (1982), Rao (1973), and Roy (1957). Nandi (1965) has shown that the test statistic obtained from Roy’s union-intersection principal is consistent if the component tests (univariate) are so, a unbiased under certain conditions, and admissible if again the component tests are admissible. We first deal with testing problems concerning means of multivariate normal populations, then we treat the case of multivariate complex
normal and that of elliptically symmetric distributions. In Section 7.3.1 we treat the problem of mean vector against one-sided alternatives for the multivariate normal populations.
