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13
Estimation of a Convex Function: Characterizations and Asymptotic Theory
, 2000
"... Abstract: We study nonparametric estimation of convex regression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case). We provide characterizations of these estimators, prove that they are consistent, and esta ..."
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Cited by 48 (20 self)
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Abstract: We study nonparametric estimation of convex regression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case). We provide characterizations of these estimators, prove that they are consistent, and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of a "invelope function" for integrated twosided Brownian motion + t 4 which is established in the companion paper Groeneboom, Jongbloed and Wellner (2001a). 1 Research supported in part by National Science Foundation grant DMS9532039 and NIAID grant 2R01 AI29196804 AMS 2000 subject classifications. Primary: 62G05; secondary 62G07, 62G08, 62E20.
Efficient Estimation for the Cox Model with Interval Censoring
 Annals of Statistics
, 1996
"... The maximum likelihood estimator (MLE) for the proportional hazards model with current status data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with p nconvergence rate and achieves the information bound, even though the MLE for the baseline cumulativ ..."
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Cited by 29 (6 self)
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The maximum likelihood estimator (MLE) for the proportional hazards model with current status data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with p nconvergence rate and achieves the information bound, even though the MLE for the baseline cumulative hazard function only converges at n 1=3 rate. Estimation of the asymptotic variance matrix for the MLE of the regression parameter is also considered. To prove our main results, we also establish a general theorem showing that the MLE of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the MLE of the infinite dimensional parameter converges at a rate slower than p n. The results are illustrated by applying them to a data set from a tumoriginicity study. 1. Introduction In many survival analysis problems, we are interested in the relationship between a failure time T and a vector of covariates Z. However, it is common that obs...
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
"... ..."
Preservation theorems for GlivenkoCantelli and uniform GlivenkoCantelli classes
 134 In High Dimensional Probability II, Evarist Giné
, 2000
"... ABSTRACT We show that the P −Glivenko property of classes of functions F1,...,Fk is preserved by a continuous function ϕ from R k to R in the sense that the new class of functions x → ϕ(f1(x),...,fk(x)), fi ∈Fi, i =1,...,k is again a GlivenkoCantelli class of functions if it has an integrable envel ..."
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Cited by 19 (8 self)
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ABSTRACT We show that the P −Glivenko property of classes of functions F1,...,Fk is preserved by a continuous function ϕ from R k to R in the sense that the new class of functions x → ϕ(f1(x),...,fk(x)), fi ∈Fi, i =1,...,k is again a GlivenkoCantelli class of functions if it has an integrable envelope. We also prove an analogous result for preservation of the uniform GlivenkoCantelli property. Corollaries of the main theorem include two preservation theorems of Dudley (1998). We apply the main result to reprove a theorem of Schick and Dudley 1998a or b? Yu (1999)concerning consistency of the NPMLE in a model for “mixed case” interval censoring. Finally a version of the consistency result of Schick and Yu (1999)is established for a general model for “mixed case interval censoring ” in which a general sample space Y is partitioned into sets which are members of some VCclass C of subsets of Y. 1 GlivenkoCantelli theorems Let (X, A,P) be a probability space, and suppose that F ⊂ L1(P). For
Consistent Estimation Of Mixture Complexity
, 2001
"... ... This article presents a semiparametric methodology yielding almost sure convergence of the estimated number of components to the true but unknown number of components. The scope of application is vast, as mixture models are routinely employed across the entire diverse application range of st ..."
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Cited by 18 (2 self)
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... This article presents a semiparametric methodology yielding almost sure convergence of the estimated number of components to the true but unknown number of components. The scope of application is vast, as mixture models are routinely employed across the entire diverse application range of statistics, including nearly all of the social and experimental sciences.
Current status data with competing risks: consistency and rates of convergence of the MLE
 Department of Statistics, University of Washington
, 2006
"... We study nonparametric estimation for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider the ‘naive estimator ’ of Jewell, Van der Laan and Henneman [10]. Groeneboom, Maathuis and Wellner [7] est ..."
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Cited by 9 (7 self)
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We study nonparametric estimation for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider the ‘naive estimator ’ of Jewell, Van der Laan and Henneman [10]. Groeneboom, Maathuis and Wellner [7] established that both estimators converge globally and locally at rate n 1/3. In this paper we use these results to derive the local limiting distributions of the estimators. The limiting distribution of the naive estimator is given by the slopes of the convex minorants of correlated Brownian motion processes with parabolic drifts. The limiting distribution of the MLE involves a new selfinduced process. We prove that this process exists and is almost surely unique. Finally, we present a simulation study showing that the MLE is superior to the naive estimator in terms of mean squared error, both for small sample sizes and asymptotically.
Misspecification in infinitedimensional Bayesian statistics
 Annals of Statistics
, 2006
"... We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P0, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the pri ..."
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Cited by 7 (0 self)
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We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P0, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the prior that minimize the Kullback–Leibler divergence with respect to P0. An entropy condition and a priormass condition determine the rate of convergence. The method is applied to several examples, with special interest for infinitedimensional models. These include Gaussian mixtures, nonparametric regression and parametric models.
Consistency of Semiparametric Maximum Likelihood Estimators for TwoPhase, Outcome Dependent Sampling
, 2000
"... this paper we consider both the absolute maximizer ( b ; b G) of the likelihood and the restricted MLE, dened through maximizing the likelihood over all pairs (; G) such that ..."
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Cited by 1 (1 self)
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this paper we consider both the absolute maximizer ( b ; b G) of the likelihood and the restricted MLE, dened through maximizing the likelihood over all pairs (; G) such that
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.
Interval Censoring
, 1994
"... Abstract The maximum likelihood estimator (MLE) for the proportional hazards model with current status data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with vnconvergence rate and achieves the information bound, even though the MLE for the baseline cum ..."
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Abstract The maximum likelihood estimator (MLE) for the proportional hazards model with current status data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with vnconvergence rate and achieves the information bound, even though the MLE for the baseline cumulative hazard function only converges at nI/3 rate. Estimation of the asymptotic variance matrix for the MLE of the regression parameter is also considered. To prove our main results, we also establish a general theorem showing that the MLE of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the MLE of the infinite dimensional parameter converges at a rate slower than vn. The results are illustrated by applying them to a data set from a tumoriginicity study. 1. Introduction In many survival ' analysis problems, we are interested in the relationship between a failure time T and a vector of covariates Z. However, it is common that observations on T are subject to censoring. Besides the familiar right censoring, many other types of censored data also arise in practice. One of them is current status data, in which it is only known whether the failure event has occurred before or after a censoring