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[wiley series in probability and statistics] methods and applications of linear models (regression and the analysis of variance) distribution of linear and quadratic forms精品.pdf

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[wiley series in probability and statistics] methods and applications of linear models (regression and the analysis of variance) distribution of linear and quadratic forms精品

Methods and Applications ofLinear Models: Regression and the Analysis of Variance,2nd Edition. Ronald R. Hocking Copyright 0 2003 John Wiley Sons, Inc. ISBN: 0-471-23222-X PART Ill : THE MATHEMATICAL THEORY OF LINEAR MODELS Methods and Applications ofLinear Models: Regression and the Analysis of Variance,2nd Edition. Ronald R. Hocking Copyright 0 2003 John Wiley Sons, Inc. ISBN: 0-471-23222-X 16 Distribution of Linear and Quadratic Forms The basic statisticsthat arise. in the analysis of linear models are either linear or quadratic functions of the data. In this chapter we develop the distributions of linear functions, and certain quadratic functions, under the assumption of normality. We also discuss the distribution ofcertain functions of these statistics that are used to construct interval estimates and tests of hypotheses on the parameters in the linear model. It is assumed that the reader has a basic knowledge of the concepts of probability distributions. The results in Part 111 are self-contained and do not depend on reading Parts I and 11; however, it will be helpfbl to read Chapter 1to motivate the concept of a linear model. 16.1 LINEAR AND QUADRATIC FORMS To motivate the discussion, we consider the following example which is one of the first problems encountered in discussions of statistical inference. h m p f e 16.1. Let pi,i = 1 , . -.,N be a random sample from the normal distributionwith mean p and variance 0’.The estimates of these parameters are given by the samplemean and variance For the purpose of making inferences on p and u2,we need the joint distri