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作者：Groeneboom, P; Jongbloed, G; Wellner, JA
作者单位：Delft University of Technology; Vrije Universiteit Amsterdam; University of Washington; University of Washington Seattle
摘要：A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively, We call this process the invelope and show that it is an almost surely uniquely defined function of integrated Brownian motion, Its role is comparable to the role of the greatest convex minorant of Brownian motion plus a parabolic drift in the problem of estimating monoto...
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作者：Klemelä, J; Tsybakov, AB
作者单位：Ruprecht Karls University Heidelberg; Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：We consider estimation of a linear functional T(f) where f is an unknown function observed in Gaussian white noise. We find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations, The results allow evaluating explicitly the effect of dimension and treating general scales of classes. Furthermore, we establish a connection between sharp adaptation and optimal recovery. Namely, we propose a scheme that reduces the construction of sharp adap...
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作者：Goldenshluger, A; Tsybakov, A
作者单位：University of Haifa; Sorbonne Universite
摘要：The problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in l(2). The method consists in an application of blockwise Stein's rule with weakly geometrically increasing blocks to the penalized least squares fits of the first N coefficients. To prove the results we develop oracle inequalities for a sequence model with correl...
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作者：Groeneboom, P; Jongbloed, G; Wellner, JA
作者单位：Delft University of Technology; Vrije Universiteit Amsterdam; University of Washington; University of Washington Seattle
摘要：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 an invelope function for integrated two-sid...
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作者：Carolan, C; Dykstra, R
作者单位：University of North Carolina; East Carolina University; University of Iowa
摘要：A clean, closed form, joint density is derived for Brownian motion, its least concave majorant, and its derivative, all at the same fixed point. Some remarkable conditional and marginal distributions follow from this,joint density, For example, it is shown that the height of the least concave majorant of Brownian motion at a fixed time point has the same distribution as the distance from the Brownian motion path to its least concave majorant at the same fixed time point. Also, it is shown that...
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作者：Banerjee, M; Wellner, JA
作者单位：University of Michigan System; University of Michigan; University of Washington; University of Washington Seattle
摘要：We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function. The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data) and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion + l(2) and greatest co...
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作者：Levitz, M; Perlman, MD; Madigan, D
作者单位：University of Washington; University of Washington Seattle; Rutgers University System; Rutgers University New Brunswick
摘要：Pearl's well-known d-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces p-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson-Madigan-Perlman (AMP) alternative Markov property for chain graphs (= adicyclic graphs), which include...
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作者：Hristache, M; Juditsky, A; Polzehl, J; Spokoiny, V
作者单位：Ecole Nationale de la Statistique et de l'Analyse de l'Information (ENSAI); Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
摘要：We propose a new method of effective dimension reduction for a multi-index model which is based on iterative improvement of the family of average derivative estimates. The procedure is computationally straightforward and does not require any prior information about the structure of the underlying model. We show that in the case when the effective dimension in of the index space does not exceed 3, this space can be estimated with the rate n (-1/2) under rather mild assumptions on the model.
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作者：Gill, RD; Robins, JM
作者单位：Utrecht University; Harvard University; Harvard T.H. Chan School of Public Health
摘要：We extend Robins' theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the g-computation formula and a collection of powerful characterizations of the g-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the cov...
