Detection boundary in sparse regression
Copyright policy )Detection boundary in sparse regression
We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to the infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p>> n). The main message is that, under some conditions, the detection boundary phenomenon that has been proved for the Gaussian sequence model, extends to high-dimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the detection boundaries sometimes depend on the knowledge of the variance in a high-dimensional setting.
[SDV]Life Sciences [q-bio], Minimax procedures in statistical decision theory, DETECTION BOUNDARY, SPARSITY, Mathematics - Statistics Theory, Statistics Theory (math.ST), 519, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 62J05, 62G08, Asymptotic properties of nonparametric inference, FOS: Mathematics, 62G05, Nonparametric hypothesis testing, minimax hypothesis testing, 62G20, HIGH-DIMENSIONAL REGRESSION;DETECTION BOUNDARY;SPARSE VECTORS;SPARSITY;MINIMAX HYPOTHESIS TESTING, 62H20, 62C20, Linear regression; mixed models, HIGH-DIMENSIONAL REGRESSION, SPARSE VECTORS, sparsity, High-dimensional regression, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], statistique, sparse vectors, high-dimensional regression, [SDV] Life Sciences [q-bio], mathématiques appliquées, detection boundary, Nonparametric estimation, MINIMAX HYPOTHESIS TESTING, 62G10
[SDV]Life Sciences [q-bio], Minimax procedures in statistical decision theory, DETECTION BOUNDARY, SPARSITY, Mathematics - Statistics Theory, Statistics Theory (math.ST), 519, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 62J05, 62G08, Asymptotic properties of nonparametric inference, FOS: Mathematics, 62G05, Nonparametric hypothesis testing, minimax hypothesis testing, 62G20, HIGH-DIMENSIONAL REGRESSION;DETECTION BOUNDARY;SPARSE VECTORS;SPARSITY;MINIMAX HYPOTHESIS TESTING, 62H20, 62C20, Linear regression; mixed models, HIGH-DIMENSIONAL REGRESSION, SPARSE VECTORS, sparsity, High-dimensional regression, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], statistique, sparse vectors, high-dimensional regression, [SDV] Life Sciences [q-bio], mathématiques appliquées, detection boundary, Nonparametric estimation, MINIMAX HYPOTHESIS TESTING, 62G10
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