This article surveys bootstrap methods for producing good approximate confidence intervals. The goal is to improve by an order of magnitude upon the accuracy of the standard intervals $\hat{\theta} \pm z^{(\alpha)} \hat{\sigma}$, in a way that allows routine application even to very complicated problems. Both theory and examples are used to show how this is done. The first seven sections provide a heuristic overview of four bootstrap confidence interval procedures: $BC_a$, bootstrap-t , ABC and calibration. Sections 8 and 9 describe the theory behind these methods, and their close connection with the likelihood-based confidence interval theory developed by Barndorff-Nielsen, Cox and Reid and others.

/pdf/bootstrap-confidence-intervals-1gss5drxvv.pdf

Bootstrap Confidence Intervals

The conventional model selection criterion AIC has been applied to choose candidate models in mixed-effects models by the consideration of marginal likelihood. Vaida and Blanchard (2005) demonstrated that such a marginal AIC and its small sample correction are inappropriate when the research focus is on clusters. Correspondingly, these authors suggested to use conditional AIC. The conditional AIC is derived under the assumptions of the variance-covariance matrix or scaled variance-covariance matrix of random effects being known. We develop a general conditional AIC but without these strong assumptions. This allows Vaida and Blanchard’s conditional AIC to be applied in a wide range. Simulation studies show that the proposed method is promising.

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A note on conditional AIC for linear mixed-effects models

/pdf/estimation-of-accuracy-in-testing-5fbjnktbf9.pdf

Estimation of Accuracy in Testing

This article surveys bootstrap methods for producing good approximate condence intervals. The goal is to improve by an order of magnitude upon the accuracy of the standard intervals O z e e O , in a way that allows routine application even to very complicated problems. Both theory and examples are used to show how this is done. The rst seven sections provide a heuristic overview of four bootstrap condence interval procedures: BCa, bootstrap-t, ABC and calibration. Sections 8 and 9 describe the theory behind these methods, and their close connec- tion with the likelihood-based condence interval theory developed by Barndorff-Nielsen, Cox and Reid and others. Condence intervals have become familiar friends in the applied statistician's collection of data-analytic tools. They combine point estima- tion and hypothesis testing into a single inferen- tial statement of great intuitive appeal. Recent advances in statistical methodology allow the con- struction of highly accurate approximate condence intervals, even for very complicated probability models and elaborate data structures. This article discusses bootstrap methods for constructing such intervals in a routine, automatic way. Two distinct approaches have guided condence interval construction since the 1930's. A small cata- logue of exact intervals has been built up for special situations, like the ratio of normal means or a sin- gle binomial parameter. However, most condence intervals are approximate, with by far the favorite approximation being the standard interval

http://staff.ustc.edu.cn/~zwp/teach/Stat-Comp/Efron_Bootstrap_CIs.pdf

Bootstrap Condence Intervals

In 1993 the International Organization for Standardization (ISO), in cooperation with several other international organizations, issued Guide to the Expression of Uncertainty in Measurement in order to establish, and standardize for international use, a set of general rules for evaluation and expressing uncertainty in measurement. The ISO recommendation has been of concern to many statisticians because it appears to combine frequentist performance measures and indices of subjective distributions in a way that neither frequentists nor Bayesians can fully endorse. The purpose of this review of the ISO Guide is to describe the essential recommendations made in the Guide, then to show how these recommendations can be regarded as approximate solutions to certain frequentist and Bayesian inference problems. The framework thus provided will, hopefully, allow statisticians to develop improvements to the ISO recommendations (particularly in the approximations used), and also better communicate with the physical science researchers who will be following the ISO guidelines.

/pdf/assessing-uncertainty-in-measurement-29zs87y96l.pdf

Assessing uncertainty in measurement

In estimation of a $p$-variate normal mean with identity covariance matrix, Stein-type estimators can offer significant gains over the $\operatorname{mle}$ in terms of risk with respect to sum of squares error loss Their maximum risk is still equal to $p$, however, which will typically be their "reported loss" In this paper we consider use of data-dependent "loss estimators" Two conditions that are attractive for such a loss estimator are that it be an improved loss estimator under some scoring rule and that it have a type of frequentist validity Loss estimators with these properties are found for several of the most important Stein-type estimators One such estimator is a generalized Bayes estimator, and the corresponding loss estimator is its posterior expected loss Thus Bayesians and frequentists can potentially agree on the analysis of this problem

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K. L. Lu

Papers

Estimation of Normal Means: Frequentist Estimation of Loss

Estimated confidence procedures for multivariate normal means