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Applied statisticians use sequential regression procedures to produce a ranking of explanatory variables and, in settings of low correlations between variables and strong true effect sizes, expect that variables at the very top of this ranking are truly relevant to the response. In a regime of certain sparsity levels, however, three examples of sequential procedures--forward stepwise, the lasso, and least angle regression--are shown to include the first spurious variable unexpectedly early. We derive a rigorous, sharp prediction of the rank of the first spurious variable for these three procedures, demonstrating that the first spurious variable occurs earlier and earlier as the regression coefficients become denser. This counterintuitive phenomenon persists for statistically independent Gaussian random designs and an arbitrarily large magnitude of the true effects. We gain a better understanding of the phenomenon by identifying the underlying cause and then leverage the insights to introduce a simple visualization tool termed the double-ranking diagram to improve on sequential methods. As a byproduct of these findings, we obtain the first provable result certifying the exact equivalence between the lasso and least angle regression in the early stages of solution paths beyond orthogonal designs. This equivalence can seamlessly carry over many important model selection results concerning the lasso to least angle regression.
Assuming that data are collected sequentially from independent streams, we consider the simultaneous testing of multiple binary hypotheses under two general setups; when the number of signals (correct alternatives) is known in advance, and when we on
Regularization aims to improve prediction performance of a given statistical modeling approach by moving to a second approach which achieves worse training error but is expected to have fewer degrees of freedom, i.e., better agreement between trainin
We study the problem of high-dimensional variable selection via some two-step procedures. First we show that given some good initial estimator which is $ell_{infty}$-consistent but not necessarily variable selection consistent, we can apply the nonne
The lasso and related sparsity inducing algorithms have been the target of substantial theoretical and applied research. Correspondingly, many results are known about their behavior for a fixed or optimally chosen tuning parameter specified up to unk
This article is concerned with the Bridge Regression, which is a special family in penalized regression with penalty function $sum_{j=1}^{p}|beta_j|^q$ with $q>0$, in a linear model with linear restrictions. The proposed restricted bridge (RBRIDGE) e