# 12GENERAL LINEAR HYPOTHESIS

## 12.1 INTRODUCTION

This chapter deals with the general linear hypothesis. In a wide variety of problems the experimenter is interested in making inferences about a vector parameter. For example, he may wish to estimate the mean of a multivariate normal or to test some hypotheses concerning the mean vector. The problem of estimation can be solved, for example, by resorting to the method of maximum likelihood estimation, discussed in Section 8.7. In this chapter we restrict ourselves to the so-called linear model problems and concern ourselves mainly with problems of hypotheses testing.

In Section 12.2 we formally describe the general model and derive a test in complete generality. In the next four sections we demonstrate the power of this test by solving four important testing problems. We will need a considerable amount of linear algebra in Section 12.2.

## 12.2 GENERAL LINEAR HYPOTHESIS

A wide variety of problems of hypotheses testing can be treated under a general setup. In this section we state the general problem and derive the test statistic and its distribution. Consider the following examples.

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