By H. Charles Romesburg
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Let y1 , y2 , . . 10. 1) where X is a known N × m matrix, β is an m-vector of unknown parameters, and e is an N-vector of random error terms. It is further assumed that e has mean vector zero and variance-covariance matrix given by a diagonal matrix with diagonal terms given by σ12 , . . , σN2 . The problem is to estimate σi2 s when they may be all unequal. C. R. Rao (1970) derived the conditions on X which ensure unbiased estimability of the σi2 s. He further introduced an estimation principle, called the minimum-norm quadratic unbiased estimation (MINQUE), and showed that the estimators of Hartley et al.
34 Chapter 10. Making Inferences about Variance Components Clearly, σˆ e2 is a biased estimator since E(σˆ e2 ) = σˆ e2 (N − q)/N . 2 This is a generalization of the notion of the restricted maximum likelihood estimation of Thompson (1962) for balanced data, considered in Volume I. Patterson and Thompson (1971) extended this to the randomized block design with unequal block sizes. , L X = 0. 3) into two parts, one of which is free of α. This is achieved by adopting a transformation suggested by Patterson and Thompson (1971).
Agricultural Biol. Environ. , 1, 490–505. 10 Some General Methods for Making Inferences about Variance Components In the study of random and mixed effects models, our interest lies primarily in making inferences about the speciﬁc variance components. In this chapter, we consider some general methods for point estimation, conﬁdence intervals, and hypothesis testing for linear models involving random effects. Most of the chapter is devoted to the study of various methods of point estimation of variance components.
Cluster analysis for researches by H. Charles Romesburg