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Developments in the theory of frequentist parametric inference in recent decades have been driven largely by the desire to achieve higher-order accuracy, in particular distributional approximations that improve on first-order asymptotic theory by one or two orders of magnitude. At the same time, much methodology is specifically designed to respect key principles of parametric inference, in particular conditionality principles. Two main routes to higher-order accuracy have emerged: analytic methods based on ‘small-sample asymptotics’, and simulation, or ‘bootstrap’, approaches. It is argued here that, of these, the simulation methodology provides a simple and effective approach, which nevertheless retains finer inferential components of theory. The paper seeks to track likely developments of parametric inference, in an era dominated by the emergence of methodological problems involving complex dependences and/or high-dimensional parameters that typically exceed available data sample sizes.
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Keywords: analytic methods; ancillary statistic; bootstrap; conditionality; full exponential family; likelihood; likelihood ratio statistic; nuisance parameter; objective Bayes; signed root likelihood ratio statistic; simulation

Document Type: Invited Paper

Publication date: June 1, 2009

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