Topics in variance component estimation

Variance components estimation originated with estimating error variance in analysis of variance by equating error mean square to its expected value. This equating procedure was _then extended to random effects models, first for balanced data (for which minimum variance properties were subsequently established) and later for unbalanced data. Unfortunately, this ANOVA ·methodology yields no optimum properties (other than unbiasedness) for estimation from unbalanced data. Today it is being replaced by maximum likelihood (ML) and restricted maximum likelihood (REML) based on normality assumptions and involving nonlinear equations that have to be solved numerically. There is also minimum norm quadratic unbiased estimation (MINQUE) which is closely related to REML but with fewer advantages. ORIGINS The analysis of variance table, as developed by R.A. Fisher in the 1920s, is a well-established summary of the arithmetic for testing hypotheses using ratios of mean squares which, under normality assumptions, have F-distributions (so named by Snedecor in honor of Fisher). This arithmetic was initially designed for what are now called fixed effects models, for which the F-statistics are suited t