A Distribution-Free Theory of Nonparametric Regression

Györfi, László

doi:10.1007/b97848

Cited by 1,384 publications

(1,894 citation statements)

References 11 publications

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“…This will become evident in the proof. Putting (27) together with (24) gives a bound (Corollary 4.2) that improves known estimates (like [20,Theorem 11.3] for instance). The improvement comes from the statistical error which is now essentially K σ 2 n as soon as the mean number nν(H k ) of simulations in each H k is large enough:…”

Section: Regression On Piecewise Constant Basis Functions and Nonparamentioning

confidence: 54%

“…Thanks to the orthogonal structure of the class K, the functions m n and m n are available in closed form [20,Ch. 4]: on each set H k , k ∈ {1, .…”

Section: Regression On Piecewise Constant Basis Functions and Nonparamentioning

confidence: 99%

“…Let us recall the classical technique to estimate the error. First, estimates on the empirical risk (23) are known [20,Theorem 11.1], and usually take the form…”

Section: Regression On Piecewise Constant Basis Functions and Nonparamentioning

confidence: 99%

“…In [18], the error analysis makes use of classical regression theory [20,Theorems 11.1 and 11.3]. In this theory, error estimates are not sufficiently tight to observe impact of variance reduction.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Gobet

Turkedjiev²

2017

Stochastic Processes and their Applications

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Please cite this article as: E. Gobet, P. Turkedjiev, Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations, Stochastic Processes and their Applications (2016), http://dx.doi.org/10. 1016/j.spa.2016.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares MonteCarlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

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Section: Regression On Piecewise Constant Basis Functions and Nonparamentioning

confidence: 54%

“…Thanks to the orthogonal structure of the class K, the functions m n and m n are available in closed form [20,Ch. 4]: on each set H k , k ∈ {1, .…”

Section: Regression On Piecewise Constant Basis Functions and Nonparamentioning

confidence: 99%

“…Let us recall the classical technique to estimate the error. First, estimates on the empirical risk (23) are known [20,Theorem 11.1], and usually take the form…”

Section: Regression On Piecewise Constant Basis Functions and Nonparamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Gobet

Turkedjiev²

2017

Stochastic Processes and their Applications

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“…Extensive work in nonparametrics has extended this result to consider the consistency of Stonetype rules under various sampling processes; see, for example, [6,10] and references therein. These models focus on various dependency structures within the training data and assume that a single processor has access to the entire data stream.…”

Section: The Classical Learning Model and Our Departurementioning

confidence: 97%

Distributed Learning in Wireless Sensor Networks

Swami

Zhao

Hong

et al. 2007

Wireless Sensor Networks

106

178

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The problem of distributed or decentralized detection and estimation in applications such as wireless sensor networks has often been considered in the framework of parametric models, in which strong assumptions are made about a statistical description of nature. In certain applications, such assumptions are warranted and systems designed from these models show promise. However, in other scenarios, prior knowledge is at best vague and translating such knowledge into a statistical model is undesirable. Applications such as these pave the way for a nonparametric study of distributed detection and estimation. In this paper, we review recent work of the authors in which some elementary models for distributed learning are considered. These models are in the spirit of classical work in nonparametric statistics and are applicable to wireless sensor networks.

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Model Selection for Additive Regression in the Presence of Right‐Censoring

Brunel

Comte²

2008

Mathematical Methods in Survival Analysis, Reliability and Quality of Life

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IntroductionStatistical tools for handling regression problems when the response is censored have been developed in the last two decades. The response was often assumed to be a linear function of the covariates, but non-parametric regression models provide a very flexible method when a general relationship between covariates and response is first to be explored. In recent years, a vast literature has been devoted to non-parametric regression estimators for completely observed data. However, few methods exist under random censoring. First, Buckley and James [BUC 79] and Koul, Susarla and Van Ryzin [KOU 81] among others introduced the original idea of transforming the data to take the censoring into account, for linear regression curves. Then, Zheng [ZHE 88] proposed various classes of unbiased transformations. Dabrowska [DAB 87] and Zheng [ZHE 88] applied non-parametric methods for estimating the univariate regression curve. Later, Fan and Gijbels [FAN 94] considered a local linear approximation for the data transformed in the same way by using a variable bandwidth adaptive to the sparsity of the design points. Györfi et al. [GYÖ 02] also studied the consistency of generalized Stone's regression estimators in the censored case. Heuchenne and Van Keilegom [HEU 05] considered a nonlinear semi-parametric regression model with censored data. Park [PAR 04] extended a procedure suggested in Gross and Lai [GRO 96] to a general non-parametric model in the presence of left-truncation and right-censoring, by using B-spline developments. Chapter written by Elodie BRUNEL and Fabienne COMTE.

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A Distribution-Free Theory of Nonparametric Regression

Cited by 1,384 publications

References 11 publications

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Distributed Learning in Wireless Sensor Networks

Model Selection for Additive Regression in the Presence of Right‐Censoring

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