Covariance Statistic and a Significance Test for Selecting All Active Variables<br/>in Lasso

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Covariance Statistic and a Significance Test for Selecting All Active Variables
in Lasso

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 49167 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s):
Abstract:
Testing the significance of the predictor variables and finding an appropriate method for inference on the coefficients is an important question in the sparse linear regression setting. Covariance statistic is a new method that tries to give an answer to this question. This statistic is defined based on lasso fitted values, and when the true model is linear, this statistic has an Exp(1) asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). From classical statistics, we have known some methods like chi-squared test for testing the significance of an additional variable between two nested linear models. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate. Since the lasso builds an adaptive sequence of linear models as the tuning parameter decreases we can’t use well-known methods like chi-squared test here. But we will show that the shrinkage in lasso plays a key role that enables us to have a statistics that is tractable and asymptotically Exp(1)
Keywords:
Large Scale Inference ; Least Absolute Shrinkage and Selection Operator (LASSO) Estimator ; Linear Regression ; Hypothesis Test ; Covariance