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23
Dirichlet Prior Sieves in Finite Normal Mixtures
 Statistica Sinica
, 2002
"... Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive ..."
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Cited by 40 (1 self)
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Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive use of the popular uniform Dirichlet prior leads to an inconsistent posterior. However, a simple adjustment to the parameters in the prior induces a random probability measure that approximates the Dirichlet process and yields a posterior that is strongly consistent for the density and weakly consistent for the unknown mixing distribution. The dimension of the resulting sieve can be selected easily in practice and a simple and efficient Gibbs sampler can be used to sample the posterior of the mixing distribution. Key words and phrases: BoseEinstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.
Semiparametric estimation of a twocomponent mixture model
 Annals of Statistics
, 2006
"... Suppose that univariate data are drawn from a mixture of two distributions that are equal up to a shift parameter. Such a model is known to be nonidentifiable from a nonparametric viewpoint. However, if we assume that the unknown mixed distribution is symmetric, we obtain the identifiability of this ..."
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Cited by 15 (3 self)
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Suppose that univariate data are drawn from a mixture of two distributions that are equal up to a shift parameter. Such a model is known to be nonidentifiable from a nonparametric viewpoint. However, if we assume that the unknown mixed distribution is symmetric, we obtain the identifiability of this model, which is then defined by four unknown parameters: the mixing proportion, two location parameters and the cumulative distribution function of the symmetric mixed distribution. We propose estimators for these four parameters when no training data is available. Our estimators are shown to be strongly consistent under mild regularity assumptions and their convergence rates are studied. Their finitesample properties are illustrated by a Monte Carlo study and our method is applied to real data.
A SimulationIntensive Approach for Checking Hierarchical Models
 TEST
, 1998
"... Recent computational advances have made it feasible to fit hierarchical models in a wide range of serious applications. If one entertains a collection of such models for a given data set, the problems of model adequacy and model choice arise. We focus on the former. While model checking usually addr ..."
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Cited by 8 (0 self)
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Recent computational advances have made it feasible to fit hierarchical models in a wide range of serious applications. If one entertains a collection of such models for a given data set, the problems of model adequacy and model choice arise. We focus on the former. While model checking usually addresses the entire model specification, model failures can occur at each hierarchical stage. Such failures include outliers, mean structure errors, dispersion misspecification, and inappropriate exchangeabilities. We propose another approach which is entirely simulation based. It only requires the model specification and that, for a given data set, one be able to simulate draws from the posterior under the model. By replicating a posterior of interest using data obtained under the model we can "see" the extent of variability in such a posterior. Then, we can compare the posterior obtained under the observed data with this medley of posterior replicates to ascertain whether the former is in agr...
Nonparametric Identification and Estimation of Multivariate Mixtures
, 2008
"... This article analyzes the identifiability of kvariate, Mcomponent finite mixture models without making parametric assumptions on the component distributions. We consider the identifiability of both the number of components and the component distributions. Under the assumption of conditionally inde ..."
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Cited by 7 (1 self)
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This article analyzes the identifiability of kvariate, Mcomponent finite mixture models without making parametric assumptions on the component distributions. We consider the identifiability of both the number of components and the component distributions. Under the assumption of conditionally independent marginals that have been used in the existing literature, we reveal an important link between the number of variables (k), the number of values each variable can take, and the number of identifiable components. The number of components (M) is nonparametrically identifiable if k ≥ 2 and each element of the variables takes at least M different values. The mixing proportions and the component distributions are nonparametrically identified if k ≥ 3 and each element of the variables takes at least M different values. Our requirement on k substantially improves the existing work, which requires either k ≥ 2M − 1 or k ≥ 6M log M. The number of components is identified by the rank of a matrix constructed from the distribution function of the data. Exploiting this property, we propose a procedure to nonparametrically estimate the number of components.
Identifying Heterogeneity in Economic Choice and Selection Models Using Mixtures, working paper
, 2009
"... We show how to nonparametrically identify the distribution of heterogeneity in a general class of structural economic choice models. We state an economic property known as reducibility and prove that reducibility ensures identification. Reducibility makes verifying the identification of nonlinear mo ..."
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Cited by 5 (0 self)
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We show how to nonparametrically identify the distribution of heterogeneity in a general class of structural economic choice models. We state an economic property known as reducibility and prove that reducibility ensures identification. Reducibility makes verifying the identification of nonlinear models a straightforward task because it is a condition that is stated directly in terms of a choice model. We can allow for a nonparametric distribution over nonparametric functions of the data. We use our framework to prove identification in three classes of economic models: 1) nonparametric regressions including with endogenous regressors, 2) multinomial discrete choice including endogenous regressors as well as multiple purchases with complementarities, and 3) selection and mixed continuousdiscrete choice. Our identification strategy avoids identification at infinity. For selection, we allow for essential heterogeneity in both the selection and outcome equations and fully identify the joint distribution of outcomes.
On identifiability in capturerecapture models
"... We study the issue of identifiability of mixture models in the context of capturerecapture abundance estimation for closed populations. Such models are used to take account of individual heterogeneity in capture probabilities, but their validity was recently questioned by Link (2003) [Biometrics 59, ..."
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Cited by 4 (0 self)
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We study the issue of identifiability of mixture models in the context of capturerecapture abundance estimation for closed populations. Such models are used to take account of individual heterogeneity in capture probabilities, but their validity was recently questioned by Link (2003) [Biometrics 59, 1123–1130] on the basis of their nonidentifiability. We give a general criterion for identifiability of the mixing distribution, and apply it to establish identifiability within families of mixing distributions that are commonly used in this context, including finite and beta mixtures. Our analysis covers binomial and geometrically distributed outcomes. Key words: abundance estimation; capturerecapture; heterogeneity; identifiability; finite mixture; beta mixture. 1
Identifiability of finite mixtures of elliptical distributions
"... We present general results on the identifiability of finite mixtures of elliptical distributions under conditions on the characteristic generators or density generators. Examples include the multivariate t distribution, symmetric stable laws, exponential power and Kotz distributions. In each case, ..."
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Cited by 4 (1 self)
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We present general results on the identifiability of finite mixtures of elliptical distributions under conditions on the characteristic generators or density generators. Examples include the multivariate t distribution, symmetric stable laws, exponential power and Kotz distributions. In each case, the shape parameter is allowed to vary in the mixture, in addition to the location vector and the scatter matrix. Furthermore, we discuss the identifiability of finite mixtures of elliptical densities with generators that correspond to scale mixtures of normal distributions. Running Heading: Identifiability of finite mixtures
A new approach to fitting linear models in high dimensional spaces
, 2000
"... This thesis presents a new approach to fitting linear models, called “pace regression”, which also overcomes the dimensionality determination problem. Its optimality in minimizing the expected prediction loss is theoretically established, when the number of free parameters is infinitely large. In th ..."
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Cited by 2 (0 self)
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This thesis presents a new approach to fitting linear models, called “pace regression”, which also overcomes the dimensionality determination problem. Its optimality in minimizing the expected prediction loss is theoretically established, when the number of free parameters is infinitely large. In this sense, pace regression outperforms existing procedures for fitting linear models. Dimensionality determination, a special case of fitting linear models, turns out to be a natural byproduct. A range of simulation studies are conducted; the results support the theoretical analysis. Through the thesis, a deeper understanding is gained of the problem of fitting linear models. Many key issues are discussed. Existing procedures, namely OLS, AIC, BIC, RIC, CIC, CV(d), BS(m), RIDGE, NNGAROTTE and LASSO, are reviewed and compared, both theoretically and empirically, with the new methods. Estimating a mixing distribution is an indispensable part of pace regression. A measurebased minimum distance approach, including probability measures and nonnegative measures, is proposed, and strongly consistent estimators are produced. Of all minimum distance methods for estimating a mixing distribution, only the