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ASYMPTOTIC LIKELIHOOD APPROXIMATIONS USING A PARTIAL LAPLACE APPROXIMATION
Author(s) -
Taylor Julian D.,
Verbyla Arūnas P.
Publication year - 2006
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2006.00451.x
Subject(s) - mathematics , laplace's method , restricted maximum likelihood , marginal likelihood , laplace transform , multiplicative function , shape parameter , scale parameter , likelihood function , maximum likelihood , statistics , mathematical analysis
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.