Semiparametric estimation of a two-component mixture model

Bordes, Laurent; Mottelet, Stéphane; Vandekerkhove, Pierre

doi:10.1214/009053606000000353

Cited by 99 publications

(190 citation statements)

References 34 publications

(50 reference statements)

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“…In a classic result, Pearson (1894) showed that the component parameters are all identified under this assumption. Similar results hold for a broad class of parametric models (Frühwirth-Schnatter, 2006) and for distributions with shape restrictions, such as symmetry (Bordes et al, 2006;Hunter et al, 2007). Note that, as we discuss in the next section, our model imposes the Normality assumption on the outcome residuals (i.e., conditional on covariates) rather than on the marginal outcome distributions.…”

Section: Principal Stratificationsupporting

confidence: 65%

Compared to what? Variation in the impacts of early childhood education by alternative care type

Feller¹,

Grindal²,

Miratrix³

et al. 2016

Ann. Appl. Stat.

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Early childhood education research often compares a group of children who receive the intervention of interest to a group of children who receive care in a range of different care settings. In this paper, we estimate differential impacts of an early childhood intervention by alternative care setting, using data from the Head Start Impact Study, a large-scale randomized evaluation. To do so, we utilize a Bayesian principal stratification framework to estimate separate impacts for two types of Compliers: those children who would otherwise be in other center-based care when assigned to control and those who would otherwise be in home-based care. We find strong, positive short-term effects of Head Start on receptive vocabulary for those Compliers who would otherwise be in home-based care. By contrast, we find no meaningful impact of Head Start on vocabulary for those Compliers who would otherwise be in other center-based care. Our findings suggest that alternative care type is a potentially important source of variation in early childhood education interventions.

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Section: Principal Stratificationsupporting

confidence: 65%

Compared to what? Variation in the impacts of early childhood education by alternative care type

Feller¹,

Grindal²,

Miratrix³

et al. 2016

Ann. Appl. Stat.

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“…Recent developments in the semi-parametric context include (Hunter et al 2007), who obtain identifiability for univariate samples by imposing a symmetry restriction on the individual components of the mixture. Further work in the univariate case includes (Cruz-Medina and Hettmansperger 2004), who assume that the component distributions are unimodal and continuous (Ellis 2002;Bordes et al 2006).…”

Section: Identifiability Considerationsmentioning

confidence: 99%

A semiparametric method for clustering mixed data

Foss

Markatou

Ray³

et al. 2016

Mach Learn

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Despite the existence of a large number of clustering algorithms, clustering remains a challenging problem. As large datasets become increasingly common in a number of different domains, it is often the case that clustering algorithms must be applied to heterogeneous sets of variables, creating an acute need for robust and scalable clustering methods for mixed continuous and categorical scale data. We show that current clustering methods for mixed-type data are generally unable to equitably balance the contribution of continuous and categorical variables without strong parametric assumptions. We develop KAMILA (KAymeans for MIxed LArge data), a clustering method that addresses this fundamental problem directly. We study theoretical aspects of our method and demonstrate its effectiveness in a series of Monte Carlo simulation studies and a set of real-world applications.

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“…Another method of constructing the estimators of these parameters is proposed in [2]. An estimator of the distribution function H 1 is constructed in [2].…”

Section: Introductionmentioning

confidence: 99%

“…The estimators proposed in the papers [2,4] are rather complicated: their evaluation is based on a numerical minimization of certain functions that depend on a sample. In the simplest case considered in [4], the function mentioned above is the U statistic.…”

Section: Introductionmentioning

confidence: 99%

Estimation of mean positions and concentrations from observations of a two-component mixture of symmetric distributions

Maiboroda

2009

Theor. Probability and Math. Statist.

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Abstract. A statistician observes a sample from a mixture of two symmetric distributions that differ from one another by a shift parameter. Estimators for mean position parameters and concentrations (mixing probabilities) for both components are constructed by the method of moments. Conditions for the consistence and asymptotic normality of these estimators are obtained. The asymptotic variance (dispersion coefficient) of the estimator of the concentration is found. IntroductionLet N subjects be observed. We assume that each of them may belong to one of the two given populations. The number δ j of the population containing the subject j is unknown. For the subject j, a numeric characteristic (variable) η j is observed. Its distribution depends on the population containing the subject, namely P{η j < x} = H i (x) if δ j = i, i = 1, 2. The (a priori) probability that the subject j belongs to the first population is p = P{δ j = 1}. Therefore P{δ j = 2} = 1 − p. Thus the data (η 1 , . . . , η N ) forms a sample of independent identically distributed random variables with the distribution functionThe data with distributions of this kind are called a sample from a finite (two-component) mixture. The populations from which the subjects are drawn are called the components of the mixture, H i are the distributions of the components, and p and 1 − p are the mixing probabilities or concentrations of the components in the mixture. There are many papers devoted to the problem of estimation of distributions and concentrations of components of finite mixtures starting with [6] and [7]. Surveys of modern methods of statistics of finite mixtures can be found in [5,8]. Most of the literature deals with parametric models of distributions, since the nonparametric models of finite mixtures are, generally speaking, not identifiable (this means that the mixing probabilities may not be uniquely estimated for the nonparametric setting).To the author's knowledge, the first paper treating the nonparametric estimation for a two-component mixture of independent identically distributed observations is [3]. The problem is identifiable in [3] due to the assumption that the observed characteristic is a vector whose dimension is at least 3 and the coordinates of these vectors (in other words, the variables characterizing the observed subjects) are independent for every component of the mixture.2000 Mathematics Subject Classification. Primary 62G07; Secondary 62G20. Key words and phrases. Method of moments, a finite mixture of probability distributions, consistence, asymptotic normality, asymptotic variance.

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Semiparametric estimation of a two-component mixture model

Cited by 99 publications

References 34 publications

Compared to what? Variation in the impacts of early childhood education by alternative care type

Compared to what? Variation in the impacts of early childhood education by alternative care type

A semiparametric method for clustering mixed data

Estimation of mean positions and concentrations from observations of a two-component mixture of symmetric distributions

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