Open Access
February 2018 Power of the spacing test for least-angle regression
Jean-Marc Azaïs, Yohann De Castro, Stéphane Mourareau
Bernoulli 24(1): 465-492 (February 2018). DOI: 10.3150/16-BEJ885


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 $\alpha$. 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 $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; 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, the Pearson’s chi-squared test (goodness of fit) and a numerical testing procedure based on the maximal correlation.

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 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.


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Jean-Marc Azaïs. Yohann De Castro. Stéphane Mourareau. "Power of the spacing test for least-angle regression." Bernoulli 24 (1) 465 - 492, February 2018.


Received: 1 October 2015; Revised: 1 May 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 1381.62110
MathSciNet: MR3706766
Digital Object Identifier: 10.3150/16-BEJ885

Keywords: $\ell_{1}$-minimization , Hypothesis testing , power , spacing test

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
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