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16
On Bayesian analysis of mixtures with an unknown number of components
 INSTITUTE OF INTERNATIONAL ECONOMICS PROJECT ON INTERNATIONAL COMPETITION POLICY," COM/DAFFE/CLP/TD(94)42
, 1997
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Model Selection for Probabilistic Clustering Using CrossValidated Likelihood
 Statistics and Computing
, 1998
"... Crossvalidated likelihood is investigated as a tool for automatically determining the appropriate number of components (given the data) in finite mixture modelling, particularly in the context of modelbased probabilistic clustering. The conceptual framework for the crossvalidation approach to mod ..."
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Cited by 65 (4 self)
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Crossvalidated likelihood is investigated as a tool for automatically determining the appropriate number of components (given the data) in finite mixture modelling, particularly in the context of modelbased probabilistic clustering. The conceptual framework for the crossvalidation approach to model selection is direct in the sense that models are judged directly on their outofsample predictive performance. The method is applied to a wellknown clustering problem in the atmospheric science literature using historical records of upper atmosphere geopotential height in the Northern hemisphere. Crossvalidated likelihood provides strong evidence for three clusters in the data set, providing an objective confirmation of earlier results derived using nonprobabilistic clustering techniques. 1 Introduction Crossvalidation is a wellknown technique in supervised learning to select a model from a family of candidate models. Examples include selecting the best classification tree using cr...
Bayesian Statistics
 in WWW', Computing Science and Statistics
, 1989
"... ∗ Signatures are on file in the Graduate School. This dissertation presents two topics from opposite disciplines: one is from a parametric realm and the other is based on nonparametric methods. The first topic is a jackknife maximum likelihood approach to statistical model selection and the second o ..."
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Cited by 20 (0 self)
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∗ Signatures are on file in the Graduate School. This dissertation presents two topics from opposite disciplines: one is from a parametric realm and the other is based on nonparametric methods. The first topic is a jackknife maximum likelihood approach to statistical model selection and the second one is a convex hull peeling depth approach to nonparametric massive multivariate data analysis. The second topic includes simulations and applications on massive astronomical data. First, we present a model selection criterion, minimizing the KullbackLeibler distance by using the jackknife method. Various model selection methods have been developed to choose a model of minimum KullbackLiebler distance to the true model, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), Minimum description length (MDL), and Bootstrap information criterion. Likewise, the jackknife method chooses a model of minimum KullbackLeibler distance through bias reduction. This bias, which is inevitable in model
Assessing model mimicry using the parametric bootstrap
 Journal of Mathematical Psychology
, 2004
"... We present a general sampling procedure to quantify model mimicry, defined as the ability of a model to account for data generated by a competing model. This sampling procedure, called the parametric bootstrap crossfitting method (PBCM; cf. Williams (J. R. Statist. Soc. B 32 (1970) 350; Biometrics ..."
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Cited by 19 (3 self)
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We present a general sampling procedure to quantify model mimicry, defined as the ability of a model to account for data generated by a competing model. This sampling procedure, called the parametric bootstrap crossfitting method (PBCM; cf. Williams (J. R. Statist. Soc. B 32 (1970) 350; Biometrics 26 (1970) 23)), generates distributions of differences in goodnessoffit expected under each of the competing models. In the data informed version of the PBCM, the generating models have specific parameter values obtained by fitting the experimental data under consideration. The data informed difference distributions can be compared to the observed difference in goodnessoffit to allow a quantification of model adequacy. In the data uninformed version of the PBCM, the generating models have a relatively broad range of parameter values based on prior knowledge. Application of both the data informed and the data uninformed PBCM is illustrated with several examples. r 2003 Elsevier Inc. All rights reserved. 1.
Modeling and testing for heterogeneity in observed strategic behavior
 Review of Economics and Statistics
, 1998
"... Experimental data have consistently shown diversity in beliefs as well as in actions among experimental subjects. This paper presents and compares alternative behavioral econometric models for the characterization of player heterogeneity, both between subpopulations of players and within subpopulat ..."
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Cited by 17 (5 self)
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Experimental data have consistently shown diversity in beliefs as well as in actions among experimental subjects. This paper presents and compares alternative behavioral econometric models for the characterization of player heterogeneity, both between subpopulations of players and within subpopulations. In particular, two econometric models of diversity within subpopulations of players are investigated: one using a model of computational errors and the other allowing for diversity in prior beliefs around a modal prior for the subpopulation.
On Model Selection and Concavity for Finite Mixture Models (Extended Abstract)
 In Proc. of Int. Symp. on Information Theory (ISIT), available at http://www.ics.uci.edu/~icadez/publications.html
, 2000
"... ) Igor V. Cadez and Padhraic Smyth Dept. of Information and Computer Science University of California Irvine, CA 926973425, U.S.A. ficadez,smythg@ics.uci.edu 1 Introduction An important open problem in density estimation and probabilistic modelbased clustering is the issue of choosing the ..."
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Cited by 4 (0 self)
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) Igor V. Cadez and Padhraic Smyth Dept. of Information and Computer Science University of California Irvine, CA 926973425, U.S.A. ficadez,smythg@ics.uci.edu 1 Introduction An important open problem in density estimation and probabilistic modelbased clustering is the issue of choosing the optimal number of components k when fitting finite mixture densities to data sets (e.g., Titterington, Makov, and Smith, 1985; McLachlan and Basford, 1988; Banfield and Raftery, 1993; Celeux and Govaert, 1995). Optimality is usually defined as choosing the model which is closest in a KullbackLeibler (KL) sense to the true datagenerating mechanism. There are several standard approaches (in a mixture density estimation context) for generating approximate estimators of this KL distance (within an additive constant) as a function of k: penalized loglikelihood such as BIC (Schwarz, 1978; Fraley and Raftery (1998)), informationtheoretic approaches such as minimum description length (MDL) (e.g....
Easy Computation of Bayes Factors and Normalizing Constants for Mixture Models via Mixture Importance Sampling
, 2001
"... We propose a method for approximating integrated likelihoods, or posterior normalizing constants, in finite mixture models, for which analytic approximations such as the Laplace method are invalid. Integrated likelihoods are key components of Bayes factors and of the posterior model probabilities us ..."
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We propose a method for approximating integrated likelihoods, or posterior normalizing constants, in finite mixture models, for which analytic approximations such as the Laplace method are invalid. Integrated likelihoods are key components of Bayes factors and of the posterior model probabilities used in Bayesian model averaging. The method starts by formulating the model in terms of the unobserved group memberships, Z, and making these, rather than the model parameters, the variables of integration. The integral is then evaluated using importance sampling over the Z. The tricky part is choosing the importance sampling function, and we study the use of mixtures as importance sampling functions. We propose two forms of this: defensive mixture importance sampling (DMIS), and Zdistance importance sampling. We choose the parameters of the mixture adaptively, and we show how this can be done so as to approximately minimize the variance of the approximation to the integral.
Estimating Components in Finite Mixtures and Hidden Markov Models
, 2003
"... When the unobservable Markov chain in a hidden Markov model is stationary the marginal distribution of the observations is a finite mixture with the number of terms equal to the number of the states of the Markov chain. This suggests estimating the number of states of the unobservable Markov chain b ..."
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Cited by 3 (1 self)
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When the unobservable Markov chain in a hidden Markov model is stationary the marginal distribution of the observations is a finite mixture with the number of terms equal to the number of the states of the Markov chain. This suggests estimating the number of states of the unobservable Markov chain by determining the number of mixture components in the marginal distribution. We therefore present new methods for estimating the number of states in a hidden Markov model, and coincidentally the unknown number of components in a finite mixture, based on penalized quasilikelihood and generalized quasilikelihood ratio methods constructed from the marginal distribution. The procedures advocated are simple to calculate and results obtained in empirical applications indicate that they are as effective as current available methods based on the full likelihood. We show that, under fairly general regularity conditions, the methods proposed will generate strongly consistent estimates of the unknown number of states or components. Some key words: finite mixture, hidden Markov process, model selection, number of states, penalized quasilikelihood, generalized quasilikelihood ratio, strong consistency. 1
Mixed normal conditional heteroskedasticity
 Journal of Financial Econometrics
, 2004
"... Both unconditional mixednormal distributions and GARCH models with fattailed conditional distributions have been employed for modeling financial return data. We consider a mixednormal distribution coupled with a GARCHtype structure which allows for conditional variance in each of the components ..."
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Both unconditional mixednormal distributions and GARCH models with fattailed conditional distributions have been employed for modeling financial return data. We consider a mixednormal distribution coupled with a GARCHtype structure which allows for conditional variance in each of the components as well as dynamic feedback between the components. Special cases and relationships with previously proposed specifications are discussed and stationarity conditions are derived. An empirical application to NASDAQindex data indicates the appropriateness of the model class and illustrates that the approach can generate a plausible disaggregation of the conditional variance process, in which the components ’ volatility dynamics have a clearly distinct behavior that is, for example, compatible with the wellknown leverage effect.
Continuous Empirical Characteristic Function Estimation of Mixtures of Normal Parameters
"... This paper develops an efficient method for estimating the discrete mixtures of normal family based on the continuous empirical characteristic function (CECF). An iterated estimation procedure based on the closed form objective distance function is proposed to improve the estimation efficiency. The ..."
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Cited by 2 (1 self)
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This paper develops an efficient method for estimating the discrete mixtures of normal family based on the continuous empirical characteristic function (CECF). An iterated estimation procedure based on the closed form objective distance function is proposed to improve the estimation efficiency. The results from the Monte Carlo simulation reveal that the CECF estimator produces good finite sample properties. In particular, it outperforms the discrete type of methods when the maximum likelihood estimation fails to converge. An empirical example is provided for illustrative purposes.