Three Examples of Accurate Likelihood Inference

Abstract

The modern theory of likelihood inference provides improved inferences in many parametric models, with little more effort than is required for application of standard first-order theory. We outline the relevant computations, and illustrate the calculations using a dilution assay, a zero-inflated Poisson regression model, and a short time series. In each case the effect of the higher order correction can be appreciable.

The Tangent Exponential Model

“…. , n, using martingale differences or a lower triangular square root of the covariance matrix for the response (Fraser et al, 2005;Lozada-Can and Davison, 2010). It is not yet clear whether other decompositions of the covariance matrix would lead to asymptotically equivalent results.…”

Modern Likelihood‐Frequentist Inference

“…Higher order likelihood inference for this example was considered by Lozada‐Can & Davison (). An interesting aspect of this is that in order to apply the Fraser approach mentioned earlier, they needed to utilise a special ancillary based on a martingale representation of the AR(1) process, in contrast to the usual ones for independent observations, or the general ancillary based on that we have in mind here.…”

“…As is typical for major advances, it is possible in retrospect to describe the main ideas and results much more simply, which is our aim. Others have chosen, usefully, to make such exposition quite differently from this, for example Brazzale et al (), Brazzale & Davison () and Lozada‐Can & Davison (). Accompanying the text by Brazzale, Davison and Reid were software tools of a different nature than here.…”

“…Special likelihood-based procedures, modifying usual inferential approximations for much higher accuracy, have emerged in recent years; see glimpses provided by Davison (2003, Chapter 12), Brazzale & Davison (2008), Lozada-Can & Davison (2010). The performance of these methods is superb and often close to exact.…”

“…As is typical for major advances, it is possible in retrospect to describe the main ideas and results much more simply, which is our aim. Others have chosen, usefully, to make such exposition quite differently from this, for example Brazzale et al (2007), Brazzale & Davison (2008 and Lozada-Can & Davison (2010). Accompanying the text by Brazzale, Davison and Reid were software tools of a different nature than here.There are largely two limiting issues regarding the adequacy of first-order methods: (i) limited information on the interest parameter and (ii) effects of fitting nuisance parameters.…”