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SOME NONASYMPTOTIC RESULTS ON RESAMPLING IN HIGH DIMENSION, I: CONFIDENCE REGIONS
 SUBMITTED TO THE ANNALS OF STATISTICS
, 2009
"... We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality ..."
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Cited by 17 (1 self)
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We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality
On the Testability of Identification in Some Nonparametric Models with Endogeneity
, 2013
"... This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogenei ..."
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Cited by 11 (1 self)
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This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions. These conditions are typically referred to as completeness conditions. The second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions. For each of these hypothesis testing problems, we provide conditions under which any test will have power no greater than size against any alternative. In this sense, we conclude that no nontrivial tests for these hypothesis testing problems exist.
Confidence Regions, Regularization and NonParametric Regression
, 2007
"... In this paper we offer a unified approach to the problem of nonparametric regression on the unit interval. It is based on a universal, honest and nonasymptotic confidence region An which is defined by a set of linear inequalities involving the values of the functions at the design points. Interest ..."
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Cited by 4 (1 self)
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In this paper we offer a unified approach to the problem of nonparametric regression on the unit interval. It is based on a universal, honest and nonasymptotic confidence region An which is defined by a set of linear inequalities involving the values of the functions at the design points. Interest will typically centre on certain simplest functions in An where simplicity can be defined in terms of shape (number of local extremes, intervals of convexity/concavity) or smoothness (bounds on derivatives) or a combination of both. Once some form of regularization has been decided upon the confidence region can be used to provide honest nonasymptotic confidence bounds which are less informative but conceptually much simpler. Although the procedure makes no attempt to minimize any loss function such as MISE the resulting estimates have optimal rates of convergence in the supremum norm both for shape and smoothness regularization. We show that rates of convergence can be misleading even for samples of size n = 10 6 and propose a different form of asymptotics which allows model complexity to increase with sample size.
Empirical Bayesian Test of the Smoothness
, 2007
"... Abstract—In the context of adaptive nonparametric curve estimation a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes. These classes are often parametrized by a quantity representing the smoothness of the signal. It has already been realized ..."
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Abstract—In the context of adaptive nonparametric curve estimation a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes. These classes are often parametrized by a quantity representing the smoothness of the signal. It has already been realized by many that the problem of estimating the smoothness is not sensible. What can then be inferred about the smoothness? The paper attempts to answer this question. We consider implications of our results to hypothesis testing about the smoothness and smoothness classification problem. The test statistic is based on the empirical Bayes approach, i.e., it is the marginalized maximum likelihood estimator of the smoothness parameter for an appropriate prior distribution on the unknown signal. Key words: empirical Bayes approach, hypothesis testing, smoothness parameter, white noise model.
Confidence Sets for the Optimal Approximating Model  Bridging a Gap between Adaptive Point Estimation and Confidence Regions
, 2008
"... In the setting of highdimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. ..."
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In the setting of highdimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. Our construction is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ²–distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.
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"... Key words and phrases. Adaptivity, confidence sets, coupling, exponential inequality, model selection, ..."
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Key words and phrases. Adaptivity, confidence sets, coupling, exponential inequality, model selection,
c © Institute of Mathematical Statistics, 2011 ON ADAPTIVE INFERENCE AND CONFIDENCE BANDS
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