Empirical Bayes estimators for a finite Markov chain

Meshkani, M R; Billard, L.

doi:10.1093/biomet/79.1.185

Cited by 19 publications

(6 citation statements)

References 12 publications

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“…Most work using Bayesian models to estimate transition probabilities has assumed a multinomial likelihood distribution and a Dirichlet prior distribution (Lee et al 1968;Satia and Lave 1973;Ezzati 1974;Meshkani and Billard 1992;McKeigue et al 2000;Ozekici and Soyer 2003;Zhao et al 2005). The models used by Cargnoni et al (1997) and Assoudou and Essebbar (2003) assumed different prior distributions including the normal distribution and Jeffreys' prior distribution, respectively.…”

Section: Previous Literaturementioning

confidence: 99%

Skill Evaluation in Women's Volleyball

Florence¹,

Fellingham²,

Vehrs³

et al. 2008

Journal of Quantitative Analysis in Sports

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BYU ScholarsArchive CitationFlorence, Lindsay W.; Fellingham, Gilbert W.; Vehrs, Pat R.; and Mortensen, Nina P., "Skill Evaluation in Women's Volleyball" (2008). AbstractThe Brigham Young University Women's Volleyball Team recorded and rated all skills (pass, set, attack, etc.) and recorded rally outcomes (point for BYU, rally continues, point for opponent) for the entire 2006 home volleyball season. Only sequences of events occurring on BYU's side of the net were considered. Events followed one of these general patterns: serve-outcome, pass-setattack-outcome, or block-dig-set-attack-outcome. These sequences of events were assumed to be first-order Markov chains where the quality of each contact depended only on the quality of the previous contact but not explicitly on contacts further removed in the sequence. We represented these sequences in an extensive matrix of transition probabilities where the elements of the matrix were the probabilities of moving from one state to another. Each row of the count matrix, consisting of the number of times play moved from one transition state to another during the season, was assumed to have a multinomial distribution. A Dirichlet prior was formulated for each row, so posterior estimates of the transition probabilities were then available using Gibbs sampling. The different paths in the transition probability matrix were followed through the possible sequences of events at each step of the MCMC process to compute the posterior probability density that a perfect pass results in a point, a perfect set results in a point, etc. These posterior probability densities are used to address questions about skill performance in BYU Women's Volleyball.

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Section: Previous Literaturementioning

confidence: 99%

Skill Evaluation in Women's Volleyball

Florence¹,

Fellingham²,

Vehrs³

et al. 2008

Journal of Quantitative Analysis in Sports

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“…Meshkani and Billard (1992), using the basic results of Anderson and Goodman (1957), developed empirical Bayes procedures for Markov chains with complete data. In typical applications, a longitudinal sequence is observed from each of a number of different individuals or specimens, and the analysis centers on the estimation of transition probabilities.…”

Section: Introductionmentioning

confidence: 99%

An Empirical Bayes Model for Markov-Dependent Binary Sequences with Randomly Missing Observations

Cole

Lee

Whitmore

et al. 1995

Journal of the American Statistical Association

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.We develop an improved empirical Bayes estimation methodology for the analysis of two-state Markov chains observed from heterogeneous individuals. First, the two transition probabilities corresponding to each chain are assumed to be drawn from a common, bivariate distribution that has beta marginals. Second, randomly missing observations are incorporated into the likelihood for the hyperparameters by efficiently summing over all possible values for the missing observations. A likelihood ratio test is used to test for dependence between the transition probabilities. Posterior distributions for the transition probabilities are also derived, as is an approximation for the equilibrium probabilities. The proposed procedures are illustrated in a numerical example and in an analysis of longitudinal store display data. Management, McGill University, for preparing the display data set for our use. The authors express their appreciation to an associate editor and two referees for their helpful comments.Their procedure uses a prior distribution that assumes that the rows of the transition matrix are stochastically inde- pendent. Bayes decision methods for finite-state Markov chains with complete observations have been discussed by Martin (1967) and Basawa and Prakasa Rao (1980). Maximum likelihood estimation with randomly missing data was discussed by Muenz and Rubinstein (1985) and Lee (1993).In this article we consider the case where some observations in each chain are missing at random (in the terminology of Rubin 1976) and the transition probabilities are drawn from a new family of bivariate beta prior distributions. That is, we develop extensions that allow for randomly missing data and for the rows of the transition matrix to be correlated. In Section 2 we develop an empirical Bayes procedure for the complete data case, and in Section 3 we extend the results to the missing-at-random case. We demonstrate the proposed procedures in Section 4 with a numerical example and an analysis of longitudinal data for store displays promoting a food product. Finally, in Section 5 we make comments on the usefulness of the proposed procedures and possible extensions. THE COMPLETE-DATA CASEWe begin by presenting the likelihood for the transition probabilities of a single, stationary Markov chain. We then introduce the Bayesian framework using a bivariate beta prior distribution and develop the empirical Bayes estimation procedure for the case where an ensemble of chains is observed. Likelihood for the Transition ProbabilitiesLet X -(X1,...,Xm+i) be a s...

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“…Estimation of the transition probability is important in Markov chain modeling. Several methods have been proposed to estimate the transition probability, namely the maximum likelihood method and the empirical Bayes method (Meshkani and Billard 1992;Lohani et al 1998;Bickenbach and Bode 2003;Paulo et al 2005;Paulo and Pereira 2007;Mishra et al 2009;Nalbantis and Tsakiris 2009). We employed the maximum likelihood method to estimate the transition probability due to its simplicity.…”

Section: Markov Chainsmentioning

confidence: 99%

The Drought Characteristics Using the First-Order Homogeneous Markov Chain of Monthly Rainfall Data in Peninsular Malaysia

Sanusi

Jemain

Zin

et al. 2014

Water Resour Manage

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Drought characteristics are important in designing the program and measures for drought mitigation and they have been used extensively by various parties in many countries. In Malaysia, the information of drought conditions and identification of drought prone areas is very limited. This study seeked to determine the drought profiles of Peninsular Malaysia using the first order homogeneous Markov chain based on Standardized Precipitation Index (SPI) of one-month time-scale. Monthly readings of rainfall data from 35 monitoring stations in Peninsular Malaysia for the period of 1970-2008 were used in the study. The procedure involved deriving the steady-state probabilities of drought events, the mean residence time for each drought category, the mean recurrence time of drought events and the mean first passage time. Analysis results showed that the longest moderate drought occurred mostly in the northwestern region of Peninsular Malaysia, while severe drought with low duration happened frequently in the middle region. The maximum duration of the severe drought condition is 2 months with majority of the severe drought areas requiring approximately 2 to 3 months to reach a non-drought condition. These results are likely to yield important insight on how to minimise the impacts of severe drought for agriculture and to avoid decadence of the water supply in areas with higher risk of severe drought. Such information would be beneficial to the agriculture planners and water resource management.Water Resour Manage

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Empirical Bayes estimators for a finite Markov chain

Cited by 19 publications

References 12 publications

Skill Evaluation in Women's Volleyball

Skill Evaluation in Women's Volleyball

An Empirical Bayes Model for Markov-Dependent Binary Sequences with Randomly Missing Observations

The Drought Characteristics Using the First-Order Homogeneous Markov Chain of Monthly Rainfall Data in Peninsular Malaysia

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