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Distributional assumptions of growth mixture models: Implications for overextraction of latent trajectory classes
 Psychological Methods
, 2003
"... Growth mixture models are often used to determine if subgroups exist within the population that follow qualitatively distinct developmental trajectories. However, statistical theory developed for finite normal mixture models suggests that latent trajectory classes can be estimated even in the absenc ..."
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Cited by 87 (8 self)
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Growth mixture models are often used to determine if subgroups exist within the population that follow qualitatively distinct developmental trajectories. However, statistical theory developed for finite normal mixture models suggests that latent trajectory classes can be estimated even in the absence of population heterogeneity if the distribution of the repeated measures is nonnormal. By drawing on this theory, this article demonstrates that multiple trajectory classes can be estimated and appear optimal for nonnormal data even when only 1 group exists in the population. Further, the withinclass parameter estimates obtained from these models are largely uninterpretable. Significant predictive relationships may be obscured or spurious relationships identified. The implications of these results for applied research are highlighted, and future directions for quantitative developments are suggested. Over the last decade, random coefficient growth modeling has become a centerpiece of longitudinal data analysis. These models have been adopted enthusiastically by applied psychological researchers in part because they provide a more dynamic analysis of repeated measures data than do many traditional techniques. However, these methods are not ideally suited for testing theories that posit the existence of qualitatively different developmental pathways, that is, theories in which distinct developmental pathways are thought to hold within subpopulations. One widely cited theory of this type is Moffitt’s (1993) distinction between “lifecourse persistent ” and “adolescentlimited ” antisocial behavior trajectories. Moffitt’s theory is prototypical of other developmental taxonomies that have been proposed in such diverse areas as developmental psychopathology (Schulenberg,
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 54 (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.
The integration of continuous and discrete latent variable models: Potential problems and promising opportunities
 Psychological Methods
, 2004
"... Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes i ..."
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Cited by 48 (6 self)
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Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecification of the structural model, nonnormal continuous measures, and nonlinear relationships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, researchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study. In recent years, many exciting developments have taken place in structural equation modeling, but perhaps none more so than the development of structural equation models that account for unobserved popula
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
"... ..."
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 8 (6 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.
CONVERGENCE OF LATENT MIXING MEASURES IN FINITE AND INFINITE MIXTURE MODELS
, 2013
"... This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and fdivergence functionals such as Hellinge ..."
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Cited by 8 (0 self)
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This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and fdivergence functionals such as Hellinger and Kullback–Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process.
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 t ..."
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Cited by 8 (3 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...
Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small
, 2003
"... We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that the maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(n ), 0 < d < 1, where n is the sample ..."
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Cited by 7 (1 self)
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We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that the maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(n ), 0 < d < 1, where n is the sample size. 1
Identifiability of mixtures of powerseries distributions and related characterizations
 Annals of the Institute of Statistical Mathematics
, 1995
"... Abstract. The concept of the identifiability of mixtures of distributions is discussed and a sufficient condition for the identifiability of the mixture of a large class of discrete distributions, namely that of the powerseries distributions, is given. Specifically, by using probabilistic argument ..."
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Cited by 6 (0 self)
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Abstract. The concept of the identifiability of mixtures of distributions is discussed and a sufficient condition for the identifiability of the mixture of a large class of discrete distributions, namely that of the powerseries distributions, is given. Specifically, by using probabilistic arguments, an elementary and shorter proof of the LiixmannEllinghaus's (1987, Statist. Probab. Lett., 5, 375378) result is obtained. Moreover, it is shown that this result is a special case of a stronger esult connected with the Stieltjes moment problem. Some recent observations due to Singh and Vasudeva (1984, J. Indian Statist.
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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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