This paper is about computations for linear mixed models having two variances, σ2e for residuals and σ2e for random effects, though the ideas can be extended to some linear mixed models having more variances. Researchers are often interested in either the restricted (residual) likelihood RL(σ2e, σ2s) or the joint posterior π(σ2e, σ2s | y) or their logarithms. Both logRL and log π can be multimodal and computations often rely on either a general purpose optimization algorithm or MCMC, both of which can fail to find regions where the target function is high. This paper presents an alternative. Letting f stand for either RL or π, we show how to find a box B in the (σ2e, σ2s) plane such that 1. all local and global maxima of log f lie within B; 2. (Equation presented) − M for a M > 0; and prespecified M and 3. log f can be estimated to within a prespecified tolerance ε everywhere in B with no danger of missing regions where log f is large. Taken together these conditions imply that the (σ2e, σ2s) plane can be divided into two parts: B, where we know logf as accurately as we wish, and Bc, where logf is small enough to be safely ignored. We provide algorithms to find B and to evaluate log f as accurately as desired everywhere in B.
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