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38
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.
A Simple Nonparametric Estimator for the Distribution of Random Coefficients in Discrete Choice Models
, 2008
"... We propose an estimator for discrete choice models, such as the logit, with a nonparametric distribution of random coefficients. The estimator is linear regression subject to linear inequality constraints and is robust, simple to program and quick to compute compared to alternative estimators for mi ..."
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Cited by 9 (3 self)
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We propose an estimator for discrete choice models, such as the logit, with a nonparametric distribution of random coefficients. The estimator is linear regression subject to linear inequality constraints and is robust, simple to program and quick to compute compared to alternative estimators for mixture models. We discuss three methods for proving identification of the distribution of heterogeneity for any given economic model. We prove the identification of the logit mixtures model, which, surprisingly given the wide use of this model over the last 30 years, is a new result. We also derive our estimator’s nonstandard asymptotic distribution and demonstrate its excellent small sample properties in a Monte Carlo. The estimator we propose can be extended to allow for endogenous prices. The estimator can also be used to reduce the computational burden of nested fixed point methods for complex models like dynamic programming discrete choice.
The Random Coefficients Logit Model is Identified
 Journal of Econometrics
, 2011
"... The random coefficients multinomial choice logit model, also known as the mixed logit, has been widely used in empirical choice analysis for the last thirty years. We prove that the distribution of random coefficients in the multinomial logit model is nonparametrically identified. Our approach requi ..."
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Cited by 9 (1 self)
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The random coefficients multinomial choice logit model, also known as the mixed logit, has been widely used in empirical choice analysis for the last thirty years. We prove that the distribution of random coefficients in the multinomial logit model is nonparametrically identified. Our approach requires variation in product characteristics only locally and does not rely on the special regressors with large supports used in related papers. One of our two identification arguments is constructive. Both approaches may be applied to other choice models with random coefficients.
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.
Nonparametric Identification and Estimation of Random Coefficients in Nonlinear Economic Models
, 2010
"... We show how to nonparametrically identify and estimate the distribution of random coefficients that characterizes the heterogeneity among agents in a general class of economic choice models. We introduce an axiom that we term separability and prove that separability of a structural model ensures ide ..."
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Cited by 7 (2 self)
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We show how to nonparametrically identify and estimate the distribution of random coefficients that characterizes the heterogeneity among agents in a general class of economic choice models. We introduce an axiom that we term separability and prove that separability of a structural model ensures identification. Identification naturally gives rise to a nonparametric minimum distance estimator. We prove identification of distributions of utility functions in multinomial choice, distributions of labor supply responses to tax changes, and distributions of wage functions in the Roy selection model. We also reconsider the problem of endogeneity in economic choice models, leading to new results on the twostage least squares model.
2007): “Defining and estimating intervention effects for groups that will develop an auxiliary outcome
 Statistical Science
"... Abstract. It has recently become popular to define treatment effects for subsets of the target population characterized by variables not observable at the time a treatment decision is made. Characterizing and estimating such treatment effects is tricky; the most popular but naive approach inappropri ..."
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Cited by 7 (0 self)
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Abstract. It has recently become popular to define treatment effects for subsets of the target population characterized by variables not observable at the time a treatment decision is made. Characterizing and estimating such treatment effects is tricky; the most popular but naive approach inappropriately adjusts for variables affected by treatment and so is biased. We consider several appropriate ways to formalize the effects: principal stratification, stratification on a single potential auxiliary variable, stratification on an observed auxiliary variable and stratification on expected levels of auxiliary variables. We then outline identifying assumptions for each type of estimand. We evaluate the utility of these estimands and estimation procedures for decision making and understanding causal processes, contrasting them with the concepts of direct and indirect effects. We motivate our development with examples from nephrology and cancer screening, and use simulated data and real data on cancer screening to illustrate the estimation methods. Key words and phrases: Causality, direct effects, interaction, effect modification, bias, principal stratification.
BOOTSTRAPPING FINITE MIXTURE MODELS
 COMPSTAT’2004 SYMPOSIUM
, 2004
"... Finite mixture regression models are used for modelling unobserved heterogeneity in the population. However, depending on the specifications these models need not be identifiable, which is especially of concern if the parameters are interpreted. As bootstrap methods are already used as a diagnostic ..."
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Cited by 6 (4 self)
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Finite mixture regression models are used for modelling unobserved heterogeneity in the population. However, depending on the specifications these models need not be identifiable, which is especially of concern if the parameters are interpreted. As bootstrap methods are already used as a diagnostic tool for linear regression models, we investigate their use for finite mixture models. We show that bootstrapping helps in revealing identifiability problems and that parametric bootstrapping can be used for analyzing the reliability of coefficient estimates.
On the Identifiability of MixturesofExperts
 Neural Networks
, 1999
"... In mixturesofexperts (ME) models, "experts" of generalized linear models are combined, according to a set of local weights called the "gating function". The invariant transformations of the ME probability density functions include the permutations of the expert labels and the translations of the p ..."
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Cited by 5 (2 self)
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In mixturesofexperts (ME) models, "experts" of generalized linear models are combined, according to a set of local weights called the "gating function". The invariant transformations of the ME probability density functions include the permutations of the expert labels and the translations of the parameters in the gating functions. Under certain conditions, we show that the ME systems are identifiable if the experts are ordered and the gating parameters are initialized. The conditions are validated for Poisson, gamma, normal and binomial experts. KeywordsGeneralized linear models, identifiability, invariant transformations, mixturesofexperts. 1 INTRODUCTION MixturesofExperts (ME) (Jacobs et. al. 1991) and Hierarchical MixturesofExperts (HME) (Jordan and Jacobs 1994) originated from the neural network literature, and have had wide applications for examining relationships among variables [Cacciatore and Nowlan (1994), Meila and Jordan (1995), Ghahramani and Hinton (1996), Tip...