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Ancillaries and conditional inference
 Statistical Science
, 2004
"... Abstract. Sufficiency has long been regarded as the primary reduction procedure to simplify a statistical model, and the assessment of the procedure involves an implicit global repeated sampling principle. By contrast, conditional procedures are almost as old and yet appear only occasionally in the ..."
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Abstract. Sufficiency has long been regarded as the primary reduction procedure to simplify a statistical model, and the assessment of the procedure involves an implicit global repeated sampling principle. By contrast, conditional procedures are almost as old and yet appear only occasionally in the central statistical literature. Recent likelihood theory examines the form of a general large sample statistical model and finds that certain natural conditional procedures provide, in wide generality, the definitive reduction from the initial variable to a variable of the same dimension as the parameter, a variable that can be viewed as directly measuring the parameter. We begin with a discussion of two intriguing examples from the literature that compare conditional and global inference methods, and come quite extraordinarily to opposite assessments concerning the appropriateness and validity of the two approaches. We then take two simple normal examples, with and without known scaling, and progressively replace the restrictive normal location assumption by more general distributional assumptions. We find that sufficiency typically becomes inapplicable and that conditional procedures from large sample likelihood theory produce the definitive reduction for the analysis. We then examine the vector parameter case and find that the elimination of nuisance parameters requires a marginalization step, not the commonly proffered conditional calculation that is based on exponential model structure. Some general conditioning and modelling criteria are then introduced. This is followed by a survey of common ancillary examples, which are then assessed for conformity to the criteria. In turn, this leads to a discussion of the place for the global repeated sampling principle in statistical inference. It is argued that the principle in conjunction with various optimality criteria has been a primary factor in the longstanding attachment to the sufficiency approach and in the related neglect of the conditioning procedures based directly on available evidence. Key words and phrases: Ancillaries, conditional inference, inference directions, likelihood, sensitivity directions, pivotal. 1.
Local limit theory and large deviations for supercritical branching processes
 Ann. Appl. Probab
"... In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn:n ≥ 1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P(Zn = vn) as vn ր ∞, and use this to study condit ..."
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In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn:n ≥ 1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P(Zn = vn) as vn ր ∞, and use this to study conditional large deviations of {YZn:n ≥ 1}, where Yn satisfies an LDP, particularly of {Z −1 n Zn+1:n ≥ 1} conditioned on Zn ≥ vn. 1. Introduction. In
Using Sketches to Estimate Twoway and Multiway Associations
 Computational Linguistics
"... We should not have to look at the entire corpus (e.g., the Web) to know if two (or more) words are associated or not. 1 A powerful sampling technique called Sketches was originally introduced to remove duplicate Web pages. We generalize sketches to estimate contingency tables and associations, using ..."
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We should not have to look at the entire corpus (e.g., the Web) to know if two (or more) words are associated or not. 1 A powerful sampling technique called Sketches was originally introduced to remove duplicate Web pages. We generalize sketches to estimate contingency tables and associations, using a maximum likelihood estimator to find the most likely contingency table given the sample, the margins (document frequencies) and the size of the collection. The proposed method has smaller errors and more flexibility than the original sketch method. Not unsurprisingly, computational work and statistical accuracy (variance or errors) depend on sampling rate, as will be shown both theoretically and empirically. Sampling methods become more and more important with larger and larger collections. At Web scale, sampling rates as low as 10 −4 may suffice. 1
Accurate Parametric Inference for Small Samples
, 2008
"... We outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice. Although the likelihood procedures are based on analytical asymptotic approximations, the focus of this paper is not on theory ..."
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We outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice. Although the likelihood procedures are based on analytical asymptotic approximations, the focus of this paper is not on theory but on implementation and applications. Numerical illustrations are given for logistic regression, nonlinear models, and linear nonnormal models, and we describe a sampling approach for the third of these classes. In the case of logistic regression, we argue that approximations are often more appropriate than ‘exact’ procedures, even when these exist.
Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings
"... The Fisher information matrix summarizes the amount of information in the data relative to the quantities of interest. There are many applications of the information matrix in modeling, systems analysis, and estimation, including confidence region calculation, input design, prediction bounds, and “n ..."
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The Fisher information matrix summarizes the amount of information in the data relative to the quantities of interest. There are many applications of the information matrix in modeling, systems analysis, and estimation, including confidence region calculation, input design, prediction bounds, and “noninformative ” priors for Bayesian analysis. This article reviews some basic principles associated with the information matrix, presents a resamplingbased method for computing the information matrix together with some new theory related to efficient implementation, and presents some numerical results. The resamplingbased method relies on an efficient technique for estimating the Hessian matrix, introduced as part of the adaptive (“secondorder”) form of the simultaneous perturbation stochastic approximation (SPSA) optimization algorithm. Key Words: Antithetic random numbers; CramérRao bound; Hessian matrix estimation; Monte Carlo simulation; Simultaneous perturbation (SP).
Recent Developments in Bootstrap Methodology
"... Abstract. Ever since its introduction, the bootstrap has provided both a powerful set of solutions for practical statisticians, and a rich source of theoretical and methodological problems for statistics. In this article, some recent developments in bootstrap methodology are reviewed and discussed. ..."
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Abstract. Ever since its introduction, the bootstrap has provided both a powerful set of solutions for practical statisticians, and a rich source of theoretical and methodological problems for statistics. In this article, some recent developments in bootstrap methodology are reviewed and discussed. After a brief introduction to the bootstrap, we consider the following topics at varying levels of detail: the use of bootstrapping for highly accurate parametric inference; theoretical properties of nonparametric bootstrapping with unequal probabilities; subsampling and the m out of n bootstrap; bootstrap failures and remedies for superefficient estimators; recent topics in significance testing; bootstrap improvements of unstable classifiers and resampling for dependent data. The treatment is telegraphic rather than exhaustive. Key words and phrases: Bagging, bootstrap, conditional inference, empirical strength probability, parametric bootstrap, subsampling, superefficient
Ancillary statistics: A review
 Statistica Sinica
, 2010
"... Ancillary statistics, one of R. A. Fisher’s most fundamental contributions to statistical inference, are statistics whose distributions do not depend on the model parameters. However, in conjunction with some other statistics, typically the maximum likelihood estimate, they provide valuable informat ..."
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Ancillary statistics, one of R. A. Fisher’s most fundamental contributions to statistical inference, are statistics whose distributions do not depend on the model parameters. However, in conjunction with some other statistics, typically the maximum likelihood estimate, they provide valuable information about the parameters of interest. The present article is a review of some of the uses and limitations of ancillary statistics. Due to the vastness of the subject, the present account is, by no means, comprehensive. The topics selected reflect our interest, and clearly many important contributions to the subject are left out. We touch upon both exact and asymptotic inference based on ancillary statistics. The discussion includes BarndorffNielsen’s p ∗ formula, the role of ancillary statistics in the elimination of nuisance parameters, and in finding optimal estimating functions. We also discuss some approximate ancillary statistics, Bayesian ancillarity and the ancillarity paradox.
Standard Errors of Fitted Component Means of Normal Mixtures
"... this paper use consider the problem of providing standard errors of the component means in normal mixture models fitted to univariate or multivariate data by maximum likelihood via the EM algorithm. Two methods of estimation of the standard errors are considered: the standard informationbased method ..."
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this paper use consider the problem of providing standard errors of the component means in normal mixture models fitted to univariate or multivariate data by maximum likelihood via the EM algorithm. Two methods of estimation of the standard errors are considered: the standard informationbased method and the computationally intensive bootstrap method. They are compared empirically by their application to three real data sets and by a smallscale Monte Carlo experiment.
. ASYMPTOTIC AND CONDITIONAL INFERENCE SOME GENERAL CONCEPTS AND RECENT DEVELOPMENTS
"... I then edited the notes, which received a final editing by Professor Cox, who also supplied the references. ..."
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I then edited the notes, which received a final editing by Professor Cox, who also supplied the references.