Inference on derivatives of high dimensional regression function with deep neural network (NN)

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We present a significance test for any given variable in nonparametric regression with many variables {via estimating derivatives of a nonparametric function}. The test is based on the moment generating function of the partial derivative of an estimator of the regression function, where the estimator is a deep neural network whose structure is allowed to become more complex as the sample size grows. This test finds applications in model specification and variable screening for high-dimensional data. To render our test applicable to high-dimensional inputs, whose dimensions can also increase with sample size, we make the assumption that the observed high-dimensional predictors can effectively serve as proxies for certain latent, lower-dimensional predictors that are actually involved in the regression function. Additionally, we finely adjust the regression function estimator, enabling us to achieve the desired asymptotic normality under the null hypothesis, as well as consistency for any fixed scenarios and certain local alternatives.