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It turns out that the conditional distribution of y 2 given y 1 is multivariate normal with mean vector and the dispersion matrix. Proof If A is idempotent with rank r, then there exists an. We have and Thus Using the assumptions in the model the expectation can be computed as Therefore Similarly it can be shown that Solving the two equations we get the estimates The estimate is necessarily nonnegative. However, may be negative, in which case we take to be zero.

It is easy to see that and are unbiased. We put a great deal of emphasis on the generalized inverse and its applications. Partly as a personal bias, I feel that the geometric approach works well in providing an vi Preface understanding of why a result should be true but has limitations when it comes to proving the result rigorously. The first three chapters are devoted to matrix theory, linear estimation, and tests of linear hypotheses, respectively. Chapter 4 collects several results on eigenval- ues and singular values that are frequently required in statistics but usually are not proved in statistics texts.

This chapter also includes sections on principal compo- nents and canonical correlations. Chapter 5 prepares the background for a course in designs, establishing the linear model as the underlying mathematical frame- work. The sections on optimality may be useful as motivation for further reading in this research area in which there is considerable activity at present. Similarly, the last chapter tries to provide a glimpse into the richness of a topic in generalized inverses rank additivity that has many interesting applications as well. Several exercises are included, some of which are used in subsequent develop- ments.

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Hints are provided for a few exercises, whereas reference to the original source is given in some other cases. I am grateful to Professors Aloke Dey, H. Neudecker, K.