Results 1  10
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17
On combinatorial testing problems
 ANNALS OF STATISTICS
, 2009
"... We study a class of hypothesis testing problems in which, upon observing the realization of an ndimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given ..."
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We study a class of hypothesis testing problems in which, upon observing the realization of an ndimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been “contaminated, ” that is, have a mean different from zero. We establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk. The combinatorial and geometric structure of the class of sets is shown to play a crucial role. The bounds are illustrated on various examples.
Preconditioning” to comply with the irrepresentable condition
, 2012
"... Preconditioning is a technique from numerical linear algebra that can accelerate algorithms to solve systems of equations. In this paper, we demonstrate how preconditioning can circumvent a stringent assumption for sign consistency in sparse linear regression. Given X ∈ Rn×p and Y ∈ Rn that satisfy ..."
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Cited by 8 (1 self)
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Preconditioning is a technique from numerical linear algebra that can accelerate algorithms to solve systems of equations. In this paper, we demonstrate how preconditioning can circumvent a stringent assumption for sign consistency in sparse linear regression. Given X ∈ Rn×p and Y ∈ Rn that satisfy the standard regression equation, this paper demonstrates that even if the design matrix X does not satisfy the irrepresentable condition for the Lasso, the design matrix FX often does, where F ∈ Rn×n is a preconditioning matrix defined in this paper. By computing the Lasso on (FX, FY), instead of on (X, Y), the necessary assumptions on X become much less stringent. Our preconditioner F ensures that the singular values of the design matrix are either zero or one. When n ≥ p, the columns of FX are orthogonal and the preconditioner always circumvents the stringent assumptions. When p ≥ n, F projects the design matrix onto the Stiefel manifold; the rows of FX are orthogonal. We give both
Central limit theorems and multiplier bootstrap when p is much larger than n
, 2012
"... Abstract. We derive a central limit theorem for the maximum of a sum of high dimensional random vectors. More precisely, we establish conditions under which the distribution of the maximum is approximated by the maximum of a sum of the Gaussian random vectors with the same covariance matrices as th ..."
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Cited by 4 (3 self)
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Abstract. We derive a central limit theorem for the maximum of a sum of high dimensional random vectors. More precisely, we establish conditions under which the distribution of the maximum is approximated by the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. The key innovation of our result is that it applies even if the dimension of random vectors (p) is much larger than the sample size (n). In fact, the growth of p could be exponential in some fractional power of n. We also show that the distribution of the maximum of a sum of the Gaussian random vectors with unknown covariance matrices can be estimated by the distribution of the maximum of the (conditional) Gaussian process obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. We call this procedure the "multiplier bootstrap". Here too, the growth of p could be exponential in some fractional power of n. We prove that our distributional approximations, either Gaussian or conditional Gaussian, yield a highquality approximation for the distribution of the original maximum, often with at most a polynomial approximation error. These results are of interest in numerous econometric and statistical applications. In particular, we demonstrate how our central limit theorem and the multiplier bootstrap can be used for high dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All of our results contain nonasymptotic bounds on approximation errors.
Bootstrap confidence sets under model misspecification Vladimir
, 2014
"... A multiplier bootstrap procedure for construction of likelihoodbased confidence sets is considered for finite samples and a possible model misspecification. Theoretical results justify the bootstrap consistency for a small or moderate sample size and allow to control the impact of the parameter dim ..."
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Cited by 2 (0 self)
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A multiplier bootstrap procedure for construction of likelihoodbased confidence sets is considered for finite samples and a possible model misspecification. Theoretical results justify the bootstrap consistency for a small or moderate sample size and allow to control the impact of the parameter dimension p: the bootstrap approximation works if p3/n is small. The main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called Small Modeling Bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modeling bias. We illustrate the results with numerical examples for misspecified constant and logistic regressions.
BOOTSTRAP AND PERMUTATION TESTS OF INDEPENDENCE FOR POINT PROCESSES
, 2015
"... Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce nonparametric test statistics, which are rescaled general Ustatistics, whose corresponding critical values are constructed from boo ..."
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Cited by 1 (0 self)
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Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce nonparametric test statistics, which are rescaled general Ustatistics, whose corresponding critical values are constructed from bootstrap and randomization/permutation approaches, making as few assumptions as possible on the underlying distribution of the point processes. We derive general consistency results for the bootstrap and for the permutation w.r.t. Wasserstein’s metric, which induces weak convergence as well as convergence of secondorder moments. The obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size, and to be consistent against any reasonable alternative. A simulation study is performed to illustrate the derived theoretical results, and to compare the performance of our new tests with existing ones in the neuroscientific literature.
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, 2013
"... Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design effective and practical econometric methods for testing this prediction in empirical analysis. Chapter 1 develops a general nonparametric frame ..."
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Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design effective and practical econometric methods for testing this prediction in empirical analysis. Chapter 1 develops a general nonparametric framework for testing monotonicity of a regression function. Using this framework, a broad class of new tests is introduced, which gives an empirical researcher a lot of flexibility to incorporate ex ante information she might have. Chapter 1 also develops new methods for simulating critical values, which are based on the combination of a bootstrap procedure and new selection algorithms. These methods yield tests that have correct asymptotic size and are asymptotically nonconservative. It is also shown how to obtain an adaptive rate optimal test that has the best attainable rate of uniform consistency against models whose regression function has Lipschitzcontinuous firstorder derivatives and that automatically adapts to the unknown smoothness of the regression function. Simulations show that the power of the new tests in many cases significantly exceeds that
ProjectTeam select Model Selection and Statistical Learning
"... e p o r t 2009 Table of contents ..."