On maximum likelihood and restricted maximum likelihood approaches to estimation of variance components

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Abstract
Two variations of the dispersion-mean correspondence model for varince components lead to the ML and REML equations. This formulation provides for addition of a nonnegativity constraint to the computational method of T. W. Anderson (1971, 1973), an iterative procedure for obtaining ML and REML estimates, which not only assures that estimates will be within the parameter space, but contributes to the stability of the algorithm as well. Estimates of the variance components obtained in this way (which includes MINQUE) are simulated for four balanced and two unbalanced random effects designs. The effect of the nonnegativity constraint on mean squared error is compared to the alternative of replacing negative values of unconstrained estimates to zero, for both the REML and ML approaches. For the two unbalanced designs, the effect of initial starting values and the significance of iterating are also assessed. The simulation results are an extension of those reported by Rich and Brown (1979) for the REML approach alone