[wiley series in probability and statistics] methods and applications of linear models (regression and the analysis of variance) estimation and inference for linear models精品.pdf

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 17 Estimation and Inference for Linear Models The purpose of this chapter is to give a detailed development of the statistical analysis of the simple linear model under the assumption of normality. Likelihood methods are used for estimation and inference. The analysis is developed for constrained and unconstrained models and the inference includes general results for testing linear hypotheses on the parameters in either model. Special topics include estimation in partitioned models and reparameterizations of the model. 17.1 ESTIMATION OF PARAMETERS The simple linear model, allowing for constraints on the parameters and assuming normality, is given by v = X @ + e , (17.1) subject to the constraints W = g , (1 7.2) where e N N(0, a21).The design matrix Xis assumed to be N x p of rank p, and the constraint matrix C is assumed to be q x p, of rank q. The parameter vector /3 and the scalar u2are unknown and must be estimated fiom the observed data. Since the distribution of the response is specified as a function of these unknown parameters, we may employ the method of maximum likelihood for parameter estimation. Simply stated, this method chooses estimates which are most likely to have led to the observed responses. This principle is interpreted as follows: Define the likelihood function as the density function for the data, viewed as a function of the unknown parameters. The maximization of this