Discussion : “ a Significance Test for the Lasso ”
نویسنده
چکیده
We congratulate the authors for an interesting article and an innovative proposal to testing the significance of the predictor variables selected by the Lasso. There is much material for thought and exploration. Research on high-dimensional regression has been very active in recent years, but most of the efforts have so far focused on estimation. Despite the popularity of the Lasso as a variable selection technique, the problem of making valid inference for a model chosen by the Lasso is largely unsettled. The current paper pinpoints some of the challenges in making valid inference in the high-dimensional setting and presents a thought-provoking approach to address them. Following the notation used in the paper, let A be the model selected at the kth step of either the Lasso or forward stepwise regression and j be the index of the variable to be added in the next step. This paper considers the problem of testing the null hypothesis that the underlying model corresponding to the true regression coefficient vector β∗ is nested in the current selected model, that is,
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Discussion : “ a Significance Test for the Lasso ”
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1. A short description of the test procedure. We start by presenting the proposed test procedure in a slightly different form than in the paper. Let β̂(λ) := arg min 2‖y −Xβ‖2 + λ‖β‖1 be the Lasso estimator with tuning parameter equal to λ. The paper uses the Lasso path {β̂(λ) :λ > 0} to construct a test statistic for the significance of certain predictor variables. For a subset S ⊆ {1, . . . , p...
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Professors Lockhart, Taylor, Tibshirani and Tibshirani are to be congratulated for their innovative and valuable contribution to the important and timely problem of testing the significance of covariates for the Lasso. Since the invention of the Lasso in Tibshirani (1996) for variable selection, there has been a huge growing literature devoted to its theory and implementation, its extensions to...
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