BAYESIAN ANALYSIS OF RANDOM-EFFECT MODELS IN THE ANALYSIS OF VARIANCE. I. POSTERIOR DISTRIBUTION OF VARIANCE-COMPONENTS.

Thus, E(yij-_U)2 = o2 + oa2. The parameters or2 and oa2 are therefore also called variancecomponents. The problem of estimating variance-components has attracted the attention of many writers-see, for example, Bross (1950), Bulmer (1957), Bush & Anderson (1963), Crump (1946, 1951), Daniels (1939), Fisher (1935), Green (1954) and Healy (1963), etc. In most of these works, the problem has been analysed from the sampling theory point of view. A difficulty which has concerned many of these writers is the so-called 'negative estimated variance' problem. For example, using the model in (1.fi1) with the added assumption that the ai's and eij's are independent among themselves, the following unbiased estimator for

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