ASYMPTOTIC LIKELIHOOD APPROXIMATIONS USING A PARTIAL LAPLACE APPROXIMATION

Abstract

Summary Elimination of a nuisance variable is often non-trivial and may involve the evaluation of an intractable integral. One approach to evaluate these integrals is to use the Laplace approximation. This paper concentrates on a new approximation, called the partial Laplace approximation, that is useful when the integrand can be partitioned into two multiplicative disjoint functions. The technique is applied to the linear mixed model and shows that the approximate likelihood obtained can be partitioned to provide a conditional likelihood for the location parameters and a marginal likelihood for the scale parameters equivalent to restricted maximum likelihood (REML). Similarly, the partial Laplace approximation is applied to the t-distribution to obtain an approximate REML for the scale parameter. A simulation study reveals that, in comparison to maximum likelihood, the scale parameter estimates of the t-distribution obtained from the approximate REML show reduced bias.