Robust Significance Testing in Sparse and High Dimensional Linear Models

Classical statistical theory offers validity under restricted assumptions. However, in practice, it is a common approach to perform statistical analysis based on data-driven model selection [1], which guarantees none of results of classical statistical theory. Those results include hypothesis testings and confidence intervals which are useful tools of measuring fitness of models. Considering that too much information about the true model of the datasets is unknown,we are unable to perform any testing before model selection. However, we are still interested in identifying how well the model we select fits the data, which leads to the problems of testing after model selection. In this paper, we discuss the robustness in testing after model selection of the lasso. Lasso, as a relatively new estimation procedure, have not been thoroughly explored yet. Especially when working in practice, one may intend to assure that the lasso model he or she chooses is the appropriate one within the assigned significance level.In the last decades, a few papers have been working on the testing problems of the lasso. Among those papers, we choose [2] as a reference paper and prove some of the lemmas of [2] with details. The lemmas help us to understand the properties of the test statistic derived in the paper, named covariance test statistic, and its asymptotic distribution under the null hypothesis. We also determine the exact stoping time for selecting the variables during the second step of the LARS algorithm. The exact stopping time allows us to propose a new LARS algorithm that is robust to the presence of outliers, by using Kendall’s τ correlation coefficient. We mimic the successive feature of the famous LARS algorithm and use the exact stopping time to select the second explanatory variables. Additionally, we propose a new test statistic that tests whether the selected variables are contained in the support of true model. The test statistics compares the covariance between the model selected before the stopping time and the model that includes an additional feature, in terms of Kendall’s τ correlation coefficient. Furthermore, we find a connection of our new test statistic with the Wilcoxon ranked-sum test and use that connection to study its distribution properties. The analysis is complicated by the intricate dependencies present in the proposed test statistic. We conjecture that the new test statistic has asymptotically rescaled normal distribution under the null. We also design a simulation to illustrate the finite sample properties of our new test statistic. We observe that the finite sample behavior shows better stability in comparison to the existing covariance test statistic.