Journal ArticleDOI
Asymptotics in Statistics–Some Basic Concepts
TLDR
In this article, the convergence of Distri butions of Likelihood Ratio has been discussed, and the authors propose a method to construct a set of limit laws for Likelihood Ratios.Abstract:
1 Introduction.- 2 Experiments, Deficiencies, Distances v.- 2.1 Comparing Risk Functions.- 2.2 Deficiency and Distance between Experiments.- 2.3 Likelihood Ratios and Blackwell's Representation.- 2.4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios.- 2.5 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger Distances, Hellinger Transforms.- 3.3 Historical Remarks.- 4 Gaussian Shift and Poisson Experiments.- 4.1 Introduction.- 4.2 Gaussian Experiments.- 4.3 Poisson Experiments.- 4.4 Historical Remarks.- 5 Limit Laws for Likelihood Ratios.- 5.1 Introduction.- 5.2 Auxiliary Results.- 5.2.1 Lindeberg's Procedure.- 5.2.2 Levy Splittings.- 5.2.3 Paul Levy's Symmetrization Inequalities.- 5.2.4 Conditions for Shift-Compactness.- 5.2.5 A Central Limit Theorem for Infinitesimal Arrays.- 5.2.6 The Special Case of Gaussian Limits.- 5.2.7 Peano Differentiable Functions.- 5.3 Limits for Binary Experiments.- 5.4 Gaussian Limits.- 5.5 Historical Remarks.- 6 Local Asymptotic Normality.- 6.1 Introduction.- 6.2 Locally Asymptotically Quadratic Families.- 6.3 A Method of Construction of Estimates.- 6.4 Some Local Bayes Properties.- 6.5 Invariance and Regularity.- 6.6 The LAMN and LAN Conditions.- 6.7 Additional Remarks on the LAN Conditions.- 6.8 Wald's Tests and Confidence Ellipsoids.- 6.9 Possible Extensions.- 6.10 Historical Remarks.- 7 Independent, Identically Distributed Observations.- 7.1 Introduction.- 7.2 The Standard i.i.d. Case: Differentiability in Quadratic Mean.- 7.3 Some Examples.- 7.4 Some Nonparametric Considerations.- 7.5 Bounds on the Risk of Estimates.- 7.6 Some Cases Where the Number of Observations Is Random.- 7.7 Historical Remarks.- 8 On Bayes Procedures.- 8.1 Introduction.- 8.2 Bayes Procedures Behave Nicely.- 8.3 The Bernstein-von Mises Phenomenon.- 8.4 A Bernstein-von Mises Result for the i.i.d. Case.- 8.5 Bayes Procedures Behave Miserably.- 8.6 Historical Remarks.- Author Index.read more
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Gaussian model selection
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References
More filters
Journal ArticleDOI
Spike and slab variable selection: Frequentist and Bayesian strategies
TL;DR: This paper introduces a variable selection method referred to as a rescaled spike and slab model, and studies the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization.
Journal ArticleDOI
Convergence rates of posterior distributions
TL;DR: In this article, the authors consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models and give general results on the rate of convergence of the posterior measure.
Journal ArticleDOI
An information statistics approach to data stream and communication complexity
TL;DR: This work presents a new method for proving strong lower bounds in communication complexity based on the notion of the conditional information complexity of a function, and shows that it also admits a direct sum theorem.
Journal ArticleDOI
Gaussian model selection
Lucien Birgé,Pascal Massart +1 more
TL;DR: The purpose in this paper is to provide a general approach to model selection via penalization for Gaussian regression and to develop the point of view about this subject.
MonographDOI
Mathematical foundations of infinite-dimensional statistical models
Evarist Giné,Richard Nickl +1 more
TL;DR: This chapter discusses nonparametric statistical models, function spaces and approximation theory, and the minimax paradigm, which aims to provide a model for adaptive inference oflihood-based procedures.