Journal of Econometrics, ISSN 0304-4076, 06/2015, Volume 186, Issue 2, pp. 325 - 344

This paper establishes non-asymptotic oracle inequalities for the prediction error and estimation accuracy of the LASSO in stationary vector autoregressive...

High-dimensional data | Oracle inequality | VAR | Adaptive LASSO | LASSO | REGRESSION | VARIABLE SELECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SHRINKAGE | MODEL SELECTION | BRIDGE ESTIMATORS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS

High-dimensional data | Oracle inequality | VAR | Adaptive LASSO | LASSO | REGRESSION | VARIABLE SELECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SHRINKAGE | MODEL SELECTION | BRIDGE ESTIMATORS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS

Journal Article

The Annals of Statistics, ISSN 0090-5364, 8/2011, Volume 39, Issue 4, pp. 2164 - 2204

We consider the problem of estimating a sparse linear regression vector β * under a Gaussian noise model, for the purpose of both prediction and model...

Minimax | Cauchy Schwarz inequality | Linear regression | Machine learning | Eigenvalues | Matrices | Random variables | Modeling | Estimators | Oracles | Group Lasso | Group sparsity | Minimax risk | Statistical learning | Moment inequality | Oracle inequalities | Penalized least squares | REGRESSION | statistical learning | penalized least squares | DANTZIG SELECTOR | group Lasso | BIAS | minimax risk | STATISTICS & PROBABILITY | VARIABLE SELECTION | LINEAR-MODELS | moment inequality | HETEROGENEITY | RECOVERY | group sparsity | AGGREGATION | Statistics | Statistics Theory | Mathematics | 62F07 | 62J05 | 62C20

Minimax | Cauchy Schwarz inequality | Linear regression | Machine learning | Eigenvalues | Matrices | Random variables | Modeling | Estimators | Oracles | Group Lasso | Group sparsity | Minimax risk | Statistical learning | Moment inequality | Oracle inequalities | Penalized least squares | REGRESSION | statistical learning | penalized least squares | DANTZIG SELECTOR | group Lasso | BIAS | minimax risk | STATISTICS & PROBABILITY | VARIABLE SELECTION | LINEAR-MODELS | moment inequality | HETEROGENEITY | RECOVERY | group sparsity | AGGREGATION | Statistics | Statistics Theory | Mathematics | 62F07 | 62J05 | 62C20

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 04/2011, Volume 57, Issue 4, pp. 2342 - 2359

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear...

oracle inequalities and semidefinite programming | matrix completion | Noise | Measurement uncertainty | Convex optimization | Minimization | norm of random matrices | Linear matrix inequalities | Noise measurement | Sparse matrices | Compressed sensing | Dantzig selector | COMPUTER SCIENCE, INFORMATION SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Matrices | Research | Information theory

oracle inequalities and semidefinite programming | matrix completion | Noise | Measurement uncertainty | Convex optimization | Minimization | norm of random matrices | Linear matrix inequalities | Noise measurement | Sparse matrices | Compressed sensing | Dantzig selector | COMPUTER SCIENCE, INFORMATION SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Matrices | Research | Information theory

Journal Article

BERNOULLI, ISSN 1350-7265, 05/2019, Volume 25, Issue 2, pp. 1225 - 1255

The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso,...

high-dimensional regression | SPARSITY | oracle inequalities | RECOVERY | OPTIMAL RATES | MODEL SELECTION | LASSO | prediction | STATISTICS & PROBABILITY | REGRESSION SHRINKAGE | VARIABLE SELECTION | SLOPE

high-dimensional regression | SPARSITY | oracle inequalities | RECOVERY | OPTIMAL RATES | MODEL SELECTION | LASSO | prediction | STATISTICS & PROBABILITY | REGRESSION SHRINKAGE | VARIABLE SELECTION | SLOPE

Journal Article

Annals of Statistics, ISSN 0090-5364, 02/2017, Volume 45, Issue 1, pp. 316 - 354

Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of...

Networks | Inhomogeneous random graph | Oracle inequality | Sparsity | Sparse graphon | Stochastic block model | stochastic block model | oracle inequality | sparse graphon | STATISTICS & PROBABILITY | sparsity | networks | Statistics | Mathematics

Networks | Inhomogeneous random graph | Oracle inequality | Sparsity | Sparse graphon | Stochastic block model | stochastic block model | oracle inequality | sparse graphon | STATISTICS & PROBABILITY | sparsity | networks | Statistics | Mathematics

Journal Article

The Annals of Statistics, ISSN 0090-5364, 12/2006, Volume 34, Issue 6, pp. 2593 - 2656

Let be a class of measurable functions f: S → [0, 1] defined on a probability space (S, , P). Given a sample of i.i.d. random variables taking values in S with...

2004 IMS Medallion Lecture | Machine learning | Least squares | Mathematical functions | Entropy | Mathematical inequalities | Mathematical minima | Learning theory | Random variables | Estimators | Oracles | Empirical risk minimization | Rademacher complexities | Model selection | Oracle inequalities | Classification | Concentration inequalities | EMPIRICAL PROCESSES | GENERALIZATION ERROR | model selection | STATISTICS & PROBABILITY | CONVEX COMBINATIONS | classification | CLASSIFIERS | oracle inequalities | CONSISTENCY | BOUNDS | ESTIMATORS | concentration inequalities | empirical risk minimization | PENALTIES | 68T10 | 62G08 | 68T05 | 60B99 | 62H30 | 68Q32

2004 IMS Medallion Lecture | Machine learning | Least squares | Mathematical functions | Entropy | Mathematical inequalities | Mathematical minima | Learning theory | Random variables | Estimators | Oracles | Empirical risk minimization | Rademacher complexities | Model selection | Oracle inequalities | Classification | Concentration inequalities | EMPIRICAL PROCESSES | GENERALIZATION ERROR | model selection | STATISTICS & PROBABILITY | CONVEX COMBINATIONS | classification | CLASSIFIERS | oracle inequalities | CONSISTENCY | BOUNDS | ESTIMATORS | concentration inequalities | empirical risk minimization | PENALTIES | 68T10 | 62G08 | 68T05 | 60B99 | 62H30 | 68Q32

Journal Article

Electronic Journal of Statistics, ISSN 1935-7524, 2007, Volume 1, pp. 169 - 194

This paper studies oracle properties of l(1)-penalized least squares in nonparametric regression setting with random design. We show that the penalized least...

Aggregation | Dimension reduction | Sparsity | Lasso | Nonparametric regression | Mutual coherence | Adaptive estimation | Oracle inequalities | Penalized least squares | oracle inequalities | nonparametric regression | penalized least squares | mutual coherence | STATISTICS & PROBABILITY | sparsity | aggregation | dimension reduction | adaptive estimation | Probability | Mathematics

Aggregation | Dimension reduction | Sparsity | Lasso | Nonparametric regression | Mutual coherence | Adaptive estimation | Oracle inequalities | Penalized least squares | oracle inequalities | nonparametric regression | penalized least squares | mutual coherence | STATISTICS & PROBABILITY | sparsity | aggregation | dimension reduction | adaptive estimation | Probability | Mathematics

Journal Article

Neurocomputing, ISSN 0925-2312, 05/2017, Volume 239, pp. 214 - 222

We investigate properties of estimators obtained by minimization of -processes with the Lasso penalty in the high-dimensional setting. Our attention is focused...

Penalized risk minimization | Oracle inequality | High-dimensional problem | U-process | Sparse model | Machine learning | MINIMIZATION | MODEL | SELECTION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Computer science | Analysis

Penalized risk minimization | Oracle inequality | High-dimensional problem | U-process | Sparse model | Machine learning | MINIMIZATION | MODEL | SELECTION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Computer science | Analysis

Journal Article

The Annals of Statistics, ISSN 0090-5364, 6/2013, Volume 41, Issue 3, pp. 1142 - 1165

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of...

Regression coefficients | Linear regression | Invertibility | Eigenvalues | Error bounds | Mathematical inequalities | Regression analysis | Estimators | Martingales | Oracles | Regression | Survival analysis | Oracle inequality | Proportional hazards | Absolute penalty | Regularization | SPARSITY | DANTZIG SELECTOR | regularization | oracle inequality | absolute penalty | STATISTICS & PROBABILITY | VARIABLE SELECTION | LARGE-SAMPLE | NONCONCAVE PENALIZED LIKELIHOOD | CONSISTENCY | PROPORTIONAL HAZARDS MODEL | survival analysis | regression | REGRESSION-MODEL | 62G05 | 62N02

Regression coefficients | Linear regression | Invertibility | Eigenvalues | Error bounds | Mathematical inequalities | Regression analysis | Estimators | Martingales | Oracles | Regression | Survival analysis | Oracle inequality | Proportional hazards | Absolute penalty | Regularization | SPARSITY | DANTZIG SELECTOR | regularization | oracle inequality | absolute penalty | STATISTICS & PROBABILITY | VARIABLE SELECTION | LARGE-SAMPLE | NONCONCAVE PENALIZED LIKELIHOOD | CONSISTENCY | PROPORTIONAL HAZARDS MODEL | survival analysis | regression | REGRESSION-MODEL | 62G05 | 62N02

Journal Article

Electronic Journal of Probability, ISSN 1083-6489, 2015, Volume 20, p. 114

We study the dimension-free inequalities, see Talagrand [49], for non-product measures extending Marton's [39] weak transport from the Hamming distance to...

Transport inequalities | Concentration of measures | Oracle inequalities | Time series | oracle inequalities | EMPIRICAL PROCESSES | DISTANCE | transport inequalities | time series | STATISTICS & PROBABILITY | concentration of measures | CHAINS | Mathematics

Transport inequalities | Concentration of measures | Oracle inequalities | Time series | oracle inequalities | EMPIRICAL PROCESSES | DISTANCE | transport inequalities | time series | STATISTICS & PROBABILITY | concentration of measures | CHAINS | Mathematics

Journal Article

Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, 2018, pp. 1 - 45

In this paper, we consider a high-dimensional statistical estimation problem in which the number of parameters is comparable or larger than the sample size. We...

High-dimensional estimation | Oracle inequality | Penalized estimation | Low-complexity models | Exponential weighted aggregation | Statistical analysis | Estimating techniques | Estimators | Inequalities | Data loss | Complexity | Statistics | Mathematics

High-dimensional estimation | Oracle inequality | Penalized estimation | Low-complexity models | Exponential weighted aggregation | Statistical analysis | Estimating techniques | Estimators | Inequalities | Data loss | Complexity | Statistics | Mathematics

Journal Article

The Annals of Statistics, ISSN 0090-5364, 6/2011, Volume 39, Issue 3, pp. 1608 - 1632

We address the problem of density estimation with 𝕃 s -loss by selection of kernel estimators. We develop a selection procedure and derive corresponding 𝕃 s...

Density estimation | Minimax | Estimate reliability | Mathematical inequalities | Mathematical functions | Density | Estimators | Oracles | Double-struck L | Kernel estimators | Empirical process | risk | Adaptive estimation | Oracle inequalities | oracle inequalities | L-s-risk | kernel estimators | empirical process | STATISTICS & PROBABILITY | adaptive estimation | Mathematics | 62G05 | Ls-risk | 62G20

Density estimation | Minimax | Estimate reliability | Mathematical inequalities | Mathematical functions | Density | Estimators | Oracles | Double-struck L | Kernel estimators | Empirical process | risk | Adaptive estimation | Oracle inequalities | oracle inequalities | L-s-risk | kernel estimators | empirical process | STATISTICS & PROBABILITY | adaptive estimation | Mathematics | 62G05 | Ls-risk | 62G20

Journal Article

Computational Mathematics and Mathematical Physics, ISSN 0965-5425, 5/2019, Volume 59, Issue 5, pp. 836 - 841

A novel analog of Nemirovski’s proximal mirror method with an adaptive choice of constants in the minimized prox-mappings at each iteration for variational...

Computational Mathematics and Numerical Analysis | proximal method | inexact oracle | adaptive method | Mathematics | variational inequality | Hölder continuous field operator | MATHEMATICS, APPLIED | Holder continuous field operator | PHYSICS, MATHEMATICAL | Algorithms | Information management | Analysis | Methods | Resveratrol | Inequalities

Computational Mathematics and Numerical Analysis | proximal method | inexact oracle | adaptive method | Mathematics | variational inequality | Hölder continuous field operator | MATHEMATICS, APPLIED | Holder continuous field operator | PHYSICS, MATHEMATICAL | Algorithms | Information management | Analysis | Methods | Resveratrol | Inequalities

Journal Article

Annals of Statistics, ISSN 0090-5364, 2017, Volume 45, Issue 1, pp. 316 - 354

Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of...

Modeling and Simulation | Computer Science | Automatic Control Engineering

Modeling and Simulation | Computer Science | Automatic Control Engineering

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 03/2019, Volume 65, Issue 3, pp. 1452 - 1472

Many statistical estimation procedures lead to nonconvex optimization problems. Algorithms to solve these problems are often guaranteed to output a stationary...

Sociology | Linear regression | Estimation | Sharp oracle inequality | Robustness | Convex functions | sparse corrected linear regression | sparse PCA | stationary points | Optimization | sparse robust regression | REGRESSION | SPARSITY | COMPUTER SCIENCE, INFORMATION SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | CONSISTENCY | RECOVERY | GUARANTEES | VARIABLES | REGULARIZATION | Economic models | Computational geometry | Algorithms | Inequalities | Estimating techniques | Convexity

Sociology | Linear regression | Estimation | Sharp oracle inequality | Robustness | Convex functions | sparse corrected linear regression | sparse PCA | stationary points | Optimization | sparse robust regression | REGRESSION | SPARSITY | COMPUTER SCIENCE, INFORMATION SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | CONSISTENCY | RECOVERY | GUARANTEES | VARIABLES | REGULARIZATION | Economic models | Computational geometry | Algorithms | Inequalities | Estimating techniques | Convexity

Journal Article

Bernoulli, ISSN 1350-7265, 8/2009, Volume 15, Issue 3, pp. 799 - 828

Let $Y_{j}=f_{\ast}(X_{j})+\xi _{j},\quad j=1,\ldots ,n$ , where $X,X_{1},\ldots ,X_{n}$ are i.i.d. random variables in a measurable space (S, ${\cal A}$ )...

Dictionaries | Logical proofs | Mathematical constants | Coordinate systems | Mathematical vectors | Mathematical inequalities | Random variables | Regression analysis | Estimators | Oracles | Regression | Sparsity | Oracle inequalities | Dantzig selector | LARGE UNDERDETERMINED SYSTEMS | oracle inequalities | regression | RECONSTRUCTION | EQUATIONS | STATISTICS & PROBABILITY | sparsity

Dictionaries | Logical proofs | Mathematical constants | Coordinate systems | Mathematical vectors | Mathematical inequalities | Random variables | Regression analysis | Estimators | Oracles | Regression | Sparsity | Oracle inequalities | Dantzig selector | LARGE UNDERDETERMINED SYSTEMS | oracle inequalities | regression | RECONSTRUCTION | EQUATIONS | STATISTICS & PROBABILITY | sparsity

Journal Article

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Full Text
Oracle inequalities and adaptive estimation in the convolution structure density model

Annals of Statistics, ISSN 0090-5364, 02/2019, Volume 47, Issue 1, pp. 233 - 287

We study the problem of nonparametric estimation under L-p-loss, p is an element of[1, infinity), in the framework of the convolution structure density model...

Deconvolution model | Density estimation | Kernel estimators | Anisotropic Nikol’skii class | Adaptive estimation | Oracle inequality | Lp-risk | anisotropic Nikol'skii class | oracle inequality | STATISTICS & PROBABILITY | L-p-risk | RATES | DECONVOLUTION | kernel estimators | BOUNDS | ADAPTATION | CONVERGENCE | SELECTION | density estimation | adaptive estimation | Statistics | Mathematics

Deconvolution model | Density estimation | Kernel estimators | Anisotropic Nikol’skii class | Adaptive estimation | Oracle inequality | Lp-risk | anisotropic Nikol'skii class | oracle inequality | STATISTICS & PROBABILITY | L-p-risk | RATES | DECONVOLUTION | kernel estimators | BOUNDS | ADAPTATION | CONVERGENCE | SELECTION | density estimation | adaptive estimation | Statistics | Mathematics

Journal Article

Statistical Inference for Stochastic Processes, ISSN 1387-0874, 7/2018, Volume 21, Issue 2, pp. 469 - 483

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi-martingale noises...

Weighted least squares estimates | Model selection | Lévy process | Improved non-asymptotic estimation | Primary 62G08 | Probability Theory and Stochastic Processes | Mathematics | Semi-Markov process | Adaptive estimation | Asymptotic efficiency | Robust quadratic risk | Sharp oracle inequality | Non-parametric regression | Secondary 62G05 | Ornstein–Uhlenbeck process | Markov processes | Analysis | Differential equations | Statistics

Weighted least squares estimates | Model selection | Lévy process | Improved non-asymptotic estimation | Primary 62G08 | Probability Theory and Stochastic Processes | Mathematics | Semi-Markov process | Adaptive estimation | Asymptotic efficiency | Robust quadratic risk | Sharp oracle inequality | Non-parametric regression | Secondary 62G05 | Ornstein–Uhlenbeck process | Markov processes | Analysis | Differential equations | Statistics

Journal Article

The Annals of Statistics, ISSN 0090-5364, 6/2002, Volume 30, Issue 3, pp. 843 - 874

We consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations. Let a finite...

Integers | Minimax | Inverse problems | Unbiased estimators | Random variables | Data smoothing | Estimators | Ellipsoids | Probabilities | Oracles | Exact minimax constants | Model selection | Statistical inverse problems | Oracle inequalities | Adaptive curve estimation | REGRESSION | oracle inequalities | statistical inverse problems | model selection | exact minimax constants | STATISTICS & PROBABILITY | HILBERT SCALES | adaptive curve estimation | WAVELET SHRINKAGE | GENERALIZED CROSS-VALIDATION | ASYMPTOTIC OPTIMALITY | 62G05 | 62G20

Integers | Minimax | Inverse problems | Unbiased estimators | Random variables | Data smoothing | Estimators | Ellipsoids | Probabilities | Oracles | Exact minimax constants | Model selection | Statistical inverse problems | Oracle inequalities | Adaptive curve estimation | REGRESSION | oracle inequalities | statistical inverse problems | model selection | exact minimax constants | STATISTICS & PROBABILITY | HILBERT SCALES | adaptive curve estimation | WAVELET SHRINKAGE | GENERALIZED CROSS-VALIDATION | ASYMPTOTIC OPTIMALITY | 62G05 | 62G20

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 07/2010, Volume 56, Issue 7, pp. 3516 - 3522

This article considers sparse signal recovery in the presence of noise. A mutual incoherence condition which was previously used for exact recovery in the...

Dictionaries | oracle inequality | ell _{1} minimization | sparse recovery | Mathematics | Vectors | Sparse matrices | Statistics | Image reconstruction | Noise level | mutual incoherence | Gaussian noise | Signal processing | Compressed sensing | Sparse recovery | Oracle inequality | minimization | Mutual incoherence | OVERCOMPLETE REPRESENTATIONS | compressed sensing | COMPUTER SCIENCE, INFORMATION SYSTEMS | RESTRICTED ISOMETRY PROPERTY | l minimization | ENGINEERING, ELECTRICAL & ELECTRONIC | Incoherence | Gaussian | Noise | Recovery | Inequalities | Information theory

Dictionaries | oracle inequality | ell _{1} minimization | sparse recovery | Mathematics | Vectors | Sparse matrices | Statistics | Image reconstruction | Noise level | mutual incoherence | Gaussian noise | Signal processing | Compressed sensing | Sparse recovery | Oracle inequality | minimization | Mutual incoherence | OVERCOMPLETE REPRESENTATIONS | compressed sensing | COMPUTER SCIENCE, INFORMATION SYSTEMS | RESTRICTED ISOMETRY PROPERTY | l minimization | ENGINEERING, ELECTRICAL & ELECTRONIC | Incoherence | Gaussian | Noise | Recovery | Inequalities | Information theory

Journal Article

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