Test by adaptive LASSO quantile method for real-time detection of a change-point

This article proposes a test statistic based on the adaptive LASSO quantile method to detect in real-time a change in a linear model. The model can have a large number of explanatory variables and the errors don’t satisfy the classical assumptions for a statistical model. For the proposed test statistic, the asymptotic distribution under H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} is obtained and the divergence under H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_1$$\end{document} is shown. It is shown via Monte Carlo simulations, in terms of empirical sizes, of empirical powers and of stopping time detection, that the useful test statistic for applications is better than other test statistics proposed in literature. Two applications on the air pollution and in the health field data are also considered.