Valid asymptotic inference after sufficient dimension reduction in a single-index framework

Despite the explosive development of sufficient dimension reduction methods, there has been limited discussion regarding the conduct of statistical inference after dimension reduction. Dimension reduction produces sufficient predictors, but this is not the endpoint of data analysis. To ensure comprehensive analysis, confidence intervals and p-values are crucial for statistical inference. Currently, the common practice involves applying the sufficient predictors in subsequent modeling as if they were the true predictors. However, this practice often leads to overly optimistic results in statistical inference. In this paper, we demonstrate how errors resulting from dimension reduction affect the standard error of regression parameters in a two-step procedure using the von-Mises expansion. We specifically choose ordinary least squares as a dimension reduction method and employ linear and logistic regression as modeling tools. Furthermore, we investigate how to improve statistical inference when the number of covariates exceeds the sample size, utilizing the seeded regression method. We also develop asymptotic theory for seeded regression with estimated seeds in high-dimensional settings. In the numerical study, we compare coverage probabilities and the growth rate of local power before and after considering the errors induced by the dimension reduction step. Finally, we apply our approach to three real datasets, covering a variety of modeling cases.