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The support reduction algorithm for computing nonparametric function estimates in mixture models
, 2003
"... ABSTRACT. In this paper, we study an algorithm (which we call the support reduction algorithm) that can be used to compute nonparametric Mestimators in mixture models. The algorithm is compared with natural competitors in the context of convex regression and the ‘Aspect problem ’ in quantum physic ..."
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Cited by 22 (7 self)
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ABSTRACT. In this paper, we study an algorithm (which we call the support reduction algorithm) that can be used to compute nonparametric Mestimators in mixture models. The algorithm is compared with natural competitors in the context of convex regression and the ‘Aspect problem ’ in quantum physics.
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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Estimation of a kmonotone density, part 2: algorithms for computation and numerical results
, 2004
"... The iterative (2k − 1)−spline algorithm is an extension of the iterative cubic spline algorithm developed and used by Groeneboom, Jongbloed, and Wellner (2001b) to compute the Least Squares Estimator (LSE) of a nonincreasing and convex density on (0, ∞), and to find an approximation of the “invelope ..."
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Cited by 4 (3 self)
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The iterative (2k − 1)−spline algorithm is an extension of the iterative cubic spline algorithm developed and used by Groeneboom, Jongbloed, and Wellner (2001b) to compute the Least Squares Estimator (LSE) of a nonincreasing and convex density on (0, ∞), and to find an approximation of the “invelope ” of the integrated twosided Brownian motion+t 4 that is involved in the limiting distribution of both the Maximum Likelihood Estimator (MLE) and the LSE (Groeneboom, Jongbloed, and Wellner (2001a)). The iterative (2k − 1) − spline algorithm was developed to compute the LSE of a kmonotone density on (0, ∞) for any integer k>2, and also to calculate an approximation of the envelopes ( “ invelopes”) of the (k − 1)fold integral of twosided Brownian motion + (k!/(2k)!) t 2k when k is odd (even) on a finite interval [−c, c] for some fixed c>0. Existence and uniqueness of the latter processes are the subject of Balabdaoui and Wellner (2004c). To compute the MLE of a kmonotone density, another variation of the algorithm involving quadratic approximation is described. This algorithm involves the computation of a spline of degree k − 1 instead of a spline of degree 2k − 1. The principles of both algorithms are explained in detail. We also give several applications to real and artificial data.
Multilevel modeling of cognitive function in schizophrenics and their first degree relatives. Multivariate Behavioral Research 2001
"... We describe multilevel modeling of cognitive function in subjects with schizophrenia, their healthy first degree relatives and controls. The purpose of the study was to compare mean cognitive performance between the three groups after adjusting for various covariates, as well as to investigate diffe ..."
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Cited by 4 (2 self)
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We describe multilevel modeling of cognitive function in subjects with schizophrenia, their healthy first degree relatives and controls. The purpose of the study was to compare mean cognitive performance between the three groups after adjusting for various covariates, as well as to investigate differences in the variances. Multilevel models were required because subjects were nested within families and some of the measures were repeated several times on the same subject. The following four methodological issues that arose during the analysis of the data are discussed. First, when the random effects distribution was not normal, nonparametric maximum likelihood (NPML) was employed, leading to a different conclusion than the conventional multilevel model regarding one of the main study hypotheses. Second, the betweensubject (withinfamily) variance was allowed to differ between the three groups. This corresponded to the variance at level 1 or level 2 depending on whether repeated measures were analyzed. Third, a positively skewed response was analyzed using a number of different generalized linear mixed models. Finally, penalized quasilikelihood (PQL) estimates for a binomial response were compared with estimates obtained using Gaussian quadrature. A small simulation study was carried out to assess the accuracy of the latter.
Global properties of kernel estimators for mixing densities in discrete exponential family models
 Statistica Sinica
, 1996
"... Abstract: This paper concerns the global performance of modifications of the kernel estimators considered in Zhang (1995) for a mixing density function g based on a sample from f(x) = � f(xθ)g(θ)dθ under weighted L ploss, 1 ≤ p ≤ ∞, where f(xθ) is a known exponential family of density functions ..."
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Cited by 2 (1 self)
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Abstract: This paper concerns the global performance of modifications of the kernel estimators considered in Zhang (1995) for a mixing density function g based on a sample from f(x) = � f(xθ)g(θ)dθ under weighted L ploss, 1 ≤ p ≤ ∞, where f(xθ) is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive upper bounds for the rate of convergence of the kernel estimators and lower bounds for the optimal convergence rate over various smoothness classes of mixing density functions. In particular under mild conditions, it is shown that these estimators achieve the optimal rate of convergence for the negative binomial mixture and are almost optimal for the Poisson mixture. Global estimation of the mixing distribution function under weighted L ploss is also considered. Key words and phrases: Mixing density, kernel estimator, discrete exponential family, rate of convergence. 1.
THE CASE FOR EXPERIENCE RATING IN MEDICAL MALPRACTICE INSURANCE: AN EMPIRICAL EVALUATION
"... Experience rating is largely absent from medical malpractice insurance contracts. This article presents evidence that physician risk differences persist, and it develops an empirical model for experience rating with a semiparametric ..."
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Cited by 2 (0 self)
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Experience rating is largely absent from medical malpractice insurance contracts. This article presents evidence that physician risk differences persist, and it develops an empirical model for experience rating with a semiparametric
BonusMalus scales using exponential loss functions
 Bulletin of the German Society of Actuaries
, 2001
"... This paper focuses on techniques for constructing BonusMalus systems in third party liability automobile insurance. Specifically, the article presents a practical method for constructing optimal BonusMalus scales with reasonable penalties that can be commercially implemented. For this purpose, the ..."
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Cited by 1 (1 self)
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This paper focuses on techniques for constructing BonusMalus systems in third party liability automobile insurance. Specifically, the article presents a practical method for constructing optimal BonusMalus scales with reasonable penalties that can be commercially implemented. For this purpose, the symmetry between the overcharges and the undercharges reflected in the usual quadratic loss function is broken through the introduction of parametric asymmetric loss functions of exponential type. The resulting system possesses the desirable financial stability property. Key words and phrases: BonusMalus system, Markov chains, exponential loss functions 1
Short Title: Weighted Nonparametric Maximum Likelihood Estimate
, 2006
"... 1 Summary. Hierarchical models have a variety of applications, including multicenter clinical trials, local estimation of disease rates, longitudinal studies, risk assessment, and metaanalysis. In a hierarchical model, observations are sampled conditional on individual unitspecific parameters an ..."
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1 Summary. Hierarchical models have a variety of applications, including multicenter clinical trials, local estimation of disease rates, longitudinal studies, risk assessment, and metaanalysis. In a hierarchical model, observations are sampled conditional on individual unitspecific parameters and these parameters are sampled from a mixing distribution. In observational studies or nonrandomized clinical trails, observations may be biased samples from a population and heterogeneous with respect to some confounding factors. Without controlling the heterogeneity in the sample, the standard estimation of the mixing distribution may lead to inaccurate statistical inferences. In this article, we propose a weighted nonparametric maximum likelihood estimate (NPMLE) of the mixing distribution and its smoothed version via weighted smoothing by roughening. The proposed estimator reduces bias by assigning a weight to each subject in the sample. The weighted NPMLE is shown to be weighted selfconsistent therefore can be easily calculated through a recursive approach. Simulation studies were conducted to evaluate the performance of the proposed estimator. We applied this method to clinical trial data evaluating a new treatment for stress urinary incontinence.
Standard Errors for EM Estimates in Generalized Linear Models with Random Effects
, 2000
"... A procedure is derived for computing standard errors of EM estimates in generalized linear models with random effects. Quadrature formulae are used to approximate the integrals in the EM algorithm, where two different approaches are pursued: GauHermite quadrature, in case of Gaussian random effects ..."
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A procedure is derived for computing standard errors of EM estimates in generalized linear models with random effects. Quadrature formulae are used to approximate the integrals in the EM algorithm, where two different approaches are pursued: GauHermite quadrature, in case of Gaussian random effects, and nonparametric maximum likelihood estimation for an unspecified random effect distribution. An approximation of the expected Fisher information matrix is derived from an expansion of the EM estimating equations. This allows for inferential arguments based on EM estimates, as demonstrated by an example and simulations. Keywords: EM algorithm, Estimating equations, GauHermite quadrature, Mixture model, Nonparametric maximum likelihood estimation, Random effect model. Institute of Statistics