Asymptotics and the theory of inference. (English) Zbl 1042.62022

Reid, N.

Ann. Stat. 31, No. 6, 1695-1731 (2003).

From the paper: Asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable. More importantly, asymptotic analysis can also provide insight into the inference process itself, suggesting what information is available and how this information may be extracted. The development of likelihood inference over the past twenty-some years provides an illustration of the interplay between techniques of approximation and statistical theory.
We consider the insight offered by asymptotic analysis for inference based on the likelihood function. This is an area that has seen considerable development in the past twenty years, largely based on asymptotic expansions and improved approximation. Section 2 reviews the main asymptotic results in likelihood inference and mentions a number of other applications of asymptotics to areas of statistics of current interest. In Section 3 we provide additional detail on a paraticular type of approximation, that of approximating \(p\)-values in tests of significance. We emphasize here recent work of Barndorff-Nielsen and colleagues and of Fraser and Reid and colleagues. In Section 4 we discuss the gap between the theoretical development and applications with special emphasis on reviewing recent work that is aimed at narrowing this gap, and outlining work still needed.

MSC:

62F12	Asymptotic properties of parametric estimators
62F05	Asymptotic properties of parametric tests
62-02	Research exposition (monographs, survey articles) pertaining to statistics
62E20	Asymptotic distribution theory in statistics

Keywords:

ancillarity; approximation; Bayesian inference; conditioning; Laplace approximation; likelihood; matching priors; saddlepoint approximation; tail area; tangent exponential model

Software:

LISP-STAT; bootlib

References:

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