Modified Profile Likelihood Research Articles

A note on the relation between modified profile likelihood and the Cox-Reid adjusted profile likelihood

SUMMARY An adjustment to the profile likelihood proposed by Cox & Reid (1987) when the parameters are orthogonal is shown to agree with modified profile likelihood in a number of instances in which the parameters are not orthogonal. Several examples of the phenomenon are given. In the case of composite transformation models with invariant parameter as the parameter of interest, it is shown that the Cox-Reid adjustment agrees with modified profile likelihood in a large-deviation sense if Lebesgue measure is rightinvariant. Necessary and sufficient conditions for equivalence in the moderate-deviation sense of modified profile likelihood and the Cox-Reid adjusted profile likelihood are

A Note on the Difference Between Profile and Modified Profile Likelihood

A Note on the Difference Between Profile and Modified Profile Likelihood

A note on the difference between profile and modified profile likelihood

The difference between profile likelihood and modified profile likelihood depends primarily on the expected value of a certain third order derivative of the log likelihood. It is shown that in exponential family problems this derivative vanishes if the parameter of interest is a mean parameter but in general not when it is a canonical parameter. As examples some matched pair problems for exponential family models are considered

Modified Profile Likelihood in Transformed Linear Models

Modified Profile Likelihood in Transformed Linear Models

Adjustments to profile likelihood

SUMMARY Conditional and marginal likelihoods constructed from parameter-dependent functions of the data have an additional parameter dependence that changes with the choice of supporting metric for the corresponding densities. We consider constructing marginal and conditional likelihoods by using densities expressed in terms of an intrinsic choice of support measure, based on either Euclidean metric or the constant information metric. Barndorff-Nielsen's (1983) approximation to the distribution of the maximum likelihood estimate is used to approximate these conditional and marginal likelihoods. The resulting likelihoods are closely related to the modified profile likelihood of Barndorff-Nielsen (1983) and the approximate conditional likelihood of Cox & Reid (1987). The normal circle problem is examined in some detail.

Differential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models

Let $\Omega$ denote the parameter space of a statistical model and let $\mathscr{K}$ be the domain of variation of the parameter of interest. Various differential-geometric structures on $\Omega$ are considered, including the expected information metric and the $\alpha$-connections studied by Chentsov and Amari, as well as the observed information metric and the observed $\alpha$-connections introduced by Barndorff-Nielsen. Under certain conditions these geometric objects on $\Omega$ can be transferred in a canonical purely differential-geometric way to $\mathscr{K}$. The transferred objects are related to structures on $\mathscr{K}$ obtained from derivatives of pseudolikelihood functions such as the profile likelihood, the modified profile likelihood and the marginal likelihood based on an $L$-sufficient statistic (cf. Remon) when such a statistic exists. For composite transformation models it is shown that the modified profile likelihood is very close to the Laplace approximation to a certain integral representation of the marginal likelihood.

Parameter Orthogonality and Approximate Conditional Inference

SUMMARY We consider inference for a scalar parameter Ψ in the presence of one or more nuisance parameters. The nuisance parameters are required to be orthogonal to the parameter of interest, and the construction and interpretation of orthogonalized parameters is discussed in some detail. For purposes of inference we propose a likelihood ratio statistic constructed from the conditional distribution of the observations, given maximum likelihood estimates for the nuisance parameters. We consider to what extent this is preferable to the profile likelihood ratio statistic in which the likelihood function is maximized over the nuisance parameters. There are close connections to the modified profile likelihood of Barndorff-Nielsen (1983). The normal transformation model of Box and Cox (1964) is discussed as an illustration.

On a Formula for the Distribution of the Maximum Likelihood Estimator

A simple formula for the conditional distribution of the maximum likelihood estimator given a maximal ancillary statistic is discussed and exemplified. The formula is generally accurate to order O(n−θθl) or even O(n−θθ3/2), and for many important models it is, in fact, exact. After some preliminary discussion of the formula and of certain relevant aspects of likelihood, the formula is used to motivate the definition of a modified profile likelihood whose inferential properties are illustrated. The question of when the distribution formula is exact is considered, and in this connexion several new examples of exactness, including a bivariate generalization of the inverse Gaussian distribution, are adduced. The formula is shown also to be exact for arbitrary transformation models. To prove this it has been necessary to extend the basic theory of transformation models to cover the cases where the group action is not free. This extension, which appears of interest in itself, also allows of a generalization of a useful formula for the marginal likelihood for the index parameter of a composite transformation model.