Selective inference with unknown variance via the square-root LASSO

Xiaoying Tian, Joshua R. Loftus, Jonathan E. Taylor

Published 2015-04-29Version 1

There has been much recent work on inference after model selection when the noise level is known, for example in forward stepwise model selection or LASSO with an independent estimate of $\sigma$. In this work we consider the more realistic scenario of an unknown noise level and propose using the square root LASSO (also known as the scaled LASSO) to conduct selective inference without previous knowledge of $\sigma$. Applying the selective inference framework described in Fithian et al. (2014), we construct selectively valid exact tests of parameters in the selected model. We discuss regression diagnostics including tests for inclusion of variables not selected by the square root LASSO. The selective inference framework also suggests a natural pseudo-likelihood estimate of the noise level that performs better than other estimates of $\sigma$ from the square root LASSO. Based on this estimate we use a Gaussian approximation to construct confidence intervals. We also consider exact inference when holding out some data for inference in the second stage, noting the same increase in power observed in Fithian et al. (2014). We illustrate our method on an HIV drug resistance dataset and in a multi-scale change point detection problem where the number of regressors is order $n^2$.