The $\infty$-S test via regression quantile affine LASSO

Abstract

The nonparametric sign test dates back to the early 18th century with a data analysis by John Arbuthnot. It is an alternative to Gosset's more recent $t$-test for consistent differences between two sets of observations. Fisher's $F$-test is a generalization of the $t$-test to linear regression and linear null hypotheses. Only the sign test is robust to non-Gaussianity. Gutenbrunner et al. [1993] derived a version of the sign test for linear null hypotheses in the spirit of the F-test, which requires the difficult estimation of the sparsity function. We propose instead a new sign test called $\infty$-S test via the convex analysis of a point estimator that thresholds the estimate towards the null hypothesis of the test.