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: Bose-Einstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.

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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 finite-sample properties are illustrated by a Monte Carlo study and our method is applied to real data.

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...

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; capture-recapture; heterogeneity; identifiability; finite mixture; beta mixture. 1

This article analyzes the identifiability of k-variate, M-component 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.

We present general results on the identifiability of finite mixtures of elliptical distrib-utions 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

In this paper three models for data generated by different linear regression distributions with Gaussian errors are discussed: Finite mixture models with random and fixed covariates and a fixed partition model. The models are compared with respect to the adequacy of their assumptions for various data situations. The interpretation of parameters is discussed. The emphasis is put on the identifiability of the parameters. Identifiability is a necessary condition for the existence of consistent estimators. It turns out that the models treated here cause other identifiability problems than simple Gaussian mixtures. This was ignored up to now and thus there are no satisfying consistency proofs in this area. Counterexamples and sufficient conditions for identifiability are given. The identifiability concept is used for fixed partition models for the first time in this paper. The concept is generalized to "partial identifiability", i.e. identifiability of only a part of the parameters. 1. INTR...

This paper discusses the possibility of exploiting price variations that may occur during the survey period of any household expenditure survey in order to identify heterogeneous demand responses to discrete price changes. This is possible since expenditure survey contain usually a large number of observations. In this paper we illustrate the feasibility of the approach in the case of the demand for lottery tickets. The exercise is made difficult because of the sampling process, which generates our data. We propose a reduced form model of purchase, which allows identification of heterogeneous responses to changes in the rollover state (i.e. whether last week jackpot has been added to the current jackpot). We show identification given the data at hand. Our estimates imply that there is substantial heterogeneity in the population both in the normal expenditure levels and in the reaction to a jackpot rolled over.

Wasserstein distances for discrete measures and convergence in nonparametric mixture models 1