Maximum Likelihood and Restricted Maximum Likelihood Estimation for a Class of Gaussian Markov Random Fields
Maximum Likelihood and Restricted Maximum Likelihood Estimation for a Class of Gaussian Markov Random Fields

Author: Victor De Oliveira

Publisher:

Published: 2009

Total Pages: 15

ISBN-13:

DOWNLOAD EBOOK

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of their maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.

Richly Parameterized Linear Models
Richly Parameterized Linear Models

Author: James S. Hodges

Publisher: CRC Press

Published: 2016-04-19

Total Pages: 464

ISBN-13: 1439866848

DOWNLOAD EBOOK

A First Step toward a Unified Theory of Richly Parameterized Linear ModelsUsing mixed linear models to analyze data often leads to results that are mysterious, inconvenient, or wrong. Further compounding the problem, statisticians lack a cohesive resource to acquire a systematic, theory-based understanding of models with random effects.Richly Param

Gaussian Markov Random Fields
Gaussian Markov Random Fields

Author: Havard Rue

Publisher: CRC Press

Published: 2005-02-18

Total Pages: 280

ISBN-13: 0203492021

DOWNLOAD EBOOK

Gaussian Markov Random Field (GMRF) models are most widely used in spatial statistics - a very active area of research in which few up-to-date reference works are available. This is the first book on the subject that provides a unified framework of GMRFs with particular emphasis on the computational aspects. This book includes extensive case-studie

Learning Continuous Sparse Pairwise Markov Random Fields
Learning Continuous Sparse Pairwise Markov Random Fields

Author: Abhin Swapnil Shah

Publisher:

Published: 2021

Total Pages: 128

ISBN-13:

DOWNLOAD EBOOK

We consider learning a sparse pairwise Markov Random Field with continuous valued variables from i.i.d samples. We adapt the framework of generalized interaction screening objective to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise Markov Random Fields with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of generalized interaction screening objective can be interpreted as local maximum likelihood estimation. As part of our analysis, we introduce a robust variation of sparse linear regression à la Lasso, which may be of interest in its own right.

Maximum Likelihood Estimation
Maximum Likelihood Estimation

Author: Scott R. Eliason

Publisher: SAGE Publications

Published: 1993-08-09

Total Pages: 100

ISBN-13: 1506315909

DOWNLOAD EBOOK

"Maximum Likelihood Estimation. . . provides a useful introduction. . . it is clear and easy to follow with applications and graphs. . . . I consider this a very useful book. . . . well-written, with a wealth of explanation. . ." --Dougal Hutchison in Educational Research Eliason reveals to the reader the underlying logic and practice of maximum likelihood (ML) estimation by providing a general modeling framework that utilizes the tools of ML methods. This framework offers readers a flexible modeling strategy since it accommodates cases from the simplest linear models (such as the normal error regression model) to the most complex nonlinear models that link a system of endogenous and exogenous variables with non-normal distributions. Using examples to illustrate the techniques of finding ML estimators and estimates, Eliason discusses what properties are desirable in an estimator, basic techniques for finding maximum likelihood solutions, the general form of the covariance matrix for ML estimates, the sampling distribution of ML estimators; the use of ML in the normal as well as other distributions, and some useful illustrations of likelihoods.