A Statistical Perspective on Ill-Posed Inverse Problems

O’Sullivan, Finbarr

doi:10.1214/ss/1177013525

Cited by 600 publications

(329 citation statements)

References 39 publications

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“…The approach has been originally introduced by O'Sullivan (1986), but the procedure finally achieved recognition due to the paper by . The approach is numerically very handy and uncovers strong similarities to penalised quasi likelihood estimation in Generalised Linear Mixed Models, as discussed in .…”

Section: Introductionmentioning

confidence: 99%

The Use of Score Tests for Inference on Variance Components

Verbeke¹,

Molenberghs²

2003

Biometrics

203

144

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Summary Whenever inference for variance components is required, the choice between one‐sided and two‐sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two‐sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one‐sided tests, however, one‐sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two‐sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well‐chosen one‐sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.

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Section: Introductionmentioning

confidence: 99%

The Use of Score Tests for Inference on Variance Components

Verbeke¹,

Molenberghs²

2003

Biometrics

203

144

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“…Other specific statistical methods that have been employed are maximum likelihood [16], errors in variables [14], robust regression [6], nonlinear regression [6], probit analysis [49], Fourier inference [20], discrimination [43], array analysis [40,42], point processes [12,17,25,30,46], moment functions [22], inverse problems [36], bootstrap [27], and sensitivity analysis [38].…”

Section: Methodsmentioning

confidence: 99%

Statistics of Earthquakes

Brillinger

2005

Encyclopedia of Statistical Sciences

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“…If, on the other hand, the problem is ill-posed (e.g., X is not invertible), then the solution is not unique, and a combination of the above two methods (16 and 18) can be used. This yields the regularization method consisting of finding E such that: (20) is achieved (see for example, Donoho et al [25] for a nice discussion of regularization within the ME formulation.) Traditionally, the positive penalization parameter D is specified to favor small sized reconstructions, meaning that out of all possible reconstructions with a given discrepancy, those with the smallest norms are chosen.…”

Section: The General Casementioning

confidence: 99%

“…See also Gzyl and Velásquez [2], which builds upon Golan and Gzyl [18] where the synthesis was first proposed. If, in addition, the data are ill-conditioned, one often has to resort to the class of regularization methods (e.g., Hoerl and Kennard [19] O'Sullivan [20], Breiman [21], Tibshirani [22], Titterington [23], Donoho et al [24]; Besnerais et al [25]. A reference for regularization in statistics is Bickel and Li [26].…”

Section: Introductionmentioning

confidence: 99%

An Entropic Estimator for Linear Inverse Problems

Golan

Gzyl

2012

Entropy

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In this paper we examine an Information-Theoretic method for solving noisy linear inverse estimation problems which encompasses under a single framework a whole class of estimation methods. Under this framework, the prior information about the unknown parameters (when such information exists), and constraints on the parameters can be incorporated in the statement of the problem. The method builds on the basics of the maximum entropy principle and consists of transforming the original problem into an estimation of a probability density on an appropriate space naturally associated with the statement of the problem. This estimation method is generic in the sense that it provides a framework for analyzing non-normal models, it is easy to implement and is suitable for all types of inverse problems such as small and or ill-conditioned, noisy data. First order approximation, large sample properties and convergence in distribution are developed as well. Analytical examples, statistics for model comparisons and evaluations, that are inherent to this method, are discussed and complemented with explicit examples.

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A Statistical Perspective on Ill-Posed Inverse Problems

Cited by 600 publications

References 39 publications

The Use of Score Tests for Inference on Variance Components

The Use of Score Tests for Inference on Variance Components

Statistics of Earthquakes

An Entropic Estimator for Linear Inverse Problems

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