ABSTRACT. Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac-Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the spacing test to the frame of unknown noise variance.More precisely, we investigate the power of the spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level ↵. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level ↵ goes to zero, we show that the probability of getting a true positive is much greater than ↵; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the spacing test for LARS and the Pearson's chi-squared test (goodness of fit).When the noise variance is unknown, our analysis unleashes a new test statistic that can be computed in cubic time in the population size and which we refer to as the t-spacing test for LARS. The t-spacing test for LARS involves the first two knots of the LARS algorithm and we give its distribution under the null hypothesis. Interestingly, numerical experiments witness that the t-spacing test for LARS enjoys the same aforementioned properties as the spacing test for LARS.
We prove the asymptotic normality of the standardized number of crossings of a centered stationary mixing Gaussian process when both the level and the time horizon go to infinity in such a way that the expected number of crossings also goes to infinity. AMS2000 Classifications. Primary 60F05. Secondary 60G15.
Abstract. The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an underdetermined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes.Although this property is the keystone of ℓ 1 -minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.
This article introduces exact testing procedures on the mean of a Gaussian process X derived from the outcomes of 1-minimization over the space of complex valued measures. The process X can be thought as the sum of two terms: first, the convolution between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive centered Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of X and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation X and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of X in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of X is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids.MSC 2010 subject classifications: Primary 62E15, 62F03, 60G15, 62H10, 62H15; secondary 60E05, 60G10, 62J05, 94A08.
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