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Exponential Family (exponential + family)
Selected AbstractsBayesian measures of model complexity and fitJOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 4 2002David J. Spiegelhalter Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ,hat' matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis. [source] Prior knowledge processing for initial state of Kalman filterINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 3 2010E. Suzdaleva Abstract The paper deals with a specification of the prior distribution of the initial state for Kalman filter. The subjective prior knowledge, used in state estimation, can be highly uncertain. In practice, incorporation of prior knowledge contributes to a good start of the filter. The present paper proposes a methodology for selection of the initial state distribution, which enables eliciting of prior knowledge from the available expert information. The proposed methodology is based on the use of the conjugate prior distribution for models belonging to the exponential family. The normal state-space model is used for demonstrating the methodology. The paper covers processing of the prior knowledge for state estimation, available in the form of simulated data. Practical experiments demonstrate the processing of prior knowledge from the urban traffic control area, which is the main application of the research. Copyright © 2009 John Wiley & Sons, Ltd. [source] Use of Kullback,Leibler divergence for forgettingINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 10 2009Miroslav Kárný Abstract Non-symmetric Kullback,Leibler divergence (KLD) measures proximity of probability density functions (pdfs). Bernardo (Ann. Stat. 1979; 7(3):686,690) had shown its unique role in approximation of pdfs. The order of the KLD arguments is also implied by his methodological result. Functional approximation of estimation and stabilized forgetting, serving for tracking of slowly varying parameters, use the reversed order. This choice has the pragmatic motivation: recursive estimator often approximates the parametric model by a member of exponential family (EF) as it maps prior pdfs from the set of conjugate pdfs (CEF) back to the CEF. Approximations based on the KLD with the reversed order of arguments preserves this property. In the paper, the approximation performed within the CEF but with the proper order of arguments of the KLD is advocated. It is applied to the parameter tracking and performance improvements are demonstrated. This practical result is of importance for adaptive systems and opens a way for improving the functional approximation. Copyright © 2008 John Wiley & Sons, Ltd. [source] Generalized Linear Models in Family StudiesJOURNAL OF MARRIAGE AND FAMILY, Issue 4 2005Zheng WU Generalized linear models (GLMs), as defined by J. A. Nelder and R. W. M. Wedderburn (1972), unify a class of regression models for categorical, discrete, and continuous response variables. As an extension of classical linear models, GLMs provide a common body of theory and methodology for some seemingly unrelated models and procedures, such as the logistic, Poisson, and probit models, that are increasingly used in family studies. This article provides an overview of the principle and the key components of GLMs, such as the exponential family of distributions, the linear predictor, and the link function. To illustrate the application of GLMs, this article uses Canadian national survey data to build an example focusing on the number of close friends among older adults. The article concludes with a discussion of the strengths and weaknesses of GLMs. [source] The complex Bingham quartic distribution and shape analysisJOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 5 2006J. T. Kent Summary., The complex Bingham distribution was introduced by Kent as a tractable model for landmark-based shape analysis. It forms an exponential family with a sufficient statistic which is quadratic in the data. However, the distribution has too much symmetry to be widely useful. In particular, under high concentration it behaves asymptotically as a normal distribution, but where the covariance matrix is constrained to have complex symmetry. To overcome this limitation and to provide a full range of asymptotic normal behaviour, we introduce a new ,complex Bingham quartic distribution' by adding a selection of quartic terms to the log-density. In the simplest case this new distribution corresponds to Kent's FB5 -distribution. Asymptotic and saddlepoint methods are developed for the normalizing constant to facilitate maximum likelihood estimation. Examples are given to show the usefulness of this new distribution. [source] Effects of correlation and missing data on sample size estimation in longitudinal clinical trialsPHARMACEUTICAL STATISTICS: THE JOURNAL OF APPLIED STATISTICS IN THE PHARMACEUTICAL INDUSTRY, Issue 1 2010Song Zhang Abstract In longitudinal clinical trials, a common objective is to compare the rates of changes in an outcome variable between two treatment groups. Generalized estimating equation (GEE) has been widely used to examine if the rates of changes are significantly different between treatment groups due to its robustness to misspecification of the true correlation structure and randomly missing data. The sample size formula for repeated outcomes is based on the assumption of missing completely at random and a large sample approximation. A simulation study is conducted to investigate the performance of GEE sample size formula with small sample sizes, damped exponential family of correlation structure and non-ignorable missing data. Copyright © 2008 John Wiley & Sons, Ltd. 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