## Abstract

This paper is about computations for linear mixed models having two variances, σ^{2}_{e} for residuals and σ^{2}_{e} 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(σ^{2}_{e}, σ^{2}_{s}) or the joint posterior π(σ^{2}_{e}, σ^{2}_{s} | 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 (σ^{2}_{e}, σ^{2}_{s}) 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 (σ^{2}_{e}, σ^{2}_{s}) plane can be divided into two parts: B, where we know logf as accurately as we wish, and B^{c}, 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.

Original language | English (US) |
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Pages (from-to) | 2293-2323 |

Number of pages | 31 |

Journal | Electronic Journal of Statistics |

Volume | 9 |

Issue number | 2 |

DOIs | |

State | Published - Aug 19 2015 |

### Bibliographical note

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