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Likelihood Functions for Inference in the Presence of a Nuisance Parameter

Thomas A. Severini
Biometrika
Vol. 85, No. 3 (Sep., 1998), pp. 507-522
Published by: Oxford University Press on behalf of Biometrika Trust
Stable URL: http://www.jstor.org/stable/2337382
Page Count: 16
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Likelihood Functions for Inference in the Presence of a Nuisance Parameter
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Abstract

Consider inference about a scalar parameter of interest θ in the presence of a vector nuisance parameter. Inference about θ is often based on a pseudolikelihood function. In this paper, the general problem of constructing a pseudo-loglikelihood function H(θ) is considered. Conditions are given under which H has the same properties as a genuine loglikelihood function for a model without a nuisance parameter. When these conditions are satisfied to a given order of approximation, H is said to be a jth-order local loglikelihood function. The theory of local loglikelihood functions is developed and it is shown that second-order versions of these have a number of desirable properties. Several commonly used pseudolikelihood functions are studied from this point of view. One commonly used pseudolikelihood function is profile likelihood in which parameters other than θ are replaced by their maximum likelihood estimates. A second aspect of the paper is to consider the use of other empirical likelihood can be viewed as one of allocating probabilities to an n-cell contingency table so as to minimise a goodness-of-fit criterion. It is shown that, when the Cressie-Read power-divergence statistic is used as the criterion, confidence regions enjoying the same convergence rates as those found for empirical likelihood can be obtained for the entire range of values of the Cressie-Read parameter λ, including -1, maximum entropy, 0, empirical likelihood, and 1. Pearson's χ2. It is noted that, in the power-divergence family, empirical likelihood is the only member which is Bartlett-correctable. However, simulation results suggest that, for the mean, using a scaled F distribution yields more accurate coverage levels for moderate sample sizes.

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