Model-free variable importance testing with machine learning methods

In this paper, we investigate variable importance testing problems in a model-free framework. Some remarkable procedures have been developed recently. Despite their success, existing procedures suffer from a significant limitation, that is, they generally require a larger training sample and do not have the fastest possible convergence rate under alternative hypothesis. In this paper, we propose a new procedure to test variable importance. Flexible machine learning methods are adopted to estimate unknown functions. Under the null hypothesis, our proposed test statistic converges to the standard chi-squared distribution. While under local alternative hypotheses, it converges to the non-central chi-square distribution. It has non-trivial power against the local alternative hypothesis which converges to the null at the fastest possible rate. We also extend our procedure to test conditional independence. Asymptotic properties are also developed. Numerical studies and two real data examples are conducted to illustrate the performance of our proposed test statistic.