A note on sufficient dimension reduction with post dimension reduction statistical inference

Sufficient dimension reduction is a widely used tool to extract core information hidden in high-dimensional data for classifying, clustering, and predicting response variables. Various dimension reduction methods and their applications have been introduced in the past decades. Data analysis using sufficient dimension reduction involves two steps: dimension reduction and model estimation. However, when we implement the two-step modeling process, we consider the estimated sufficient predictor as a true predictor variable and proceed to the model development step, which includes statistical inference such as estimating confidence intervals and performing hypothesis tests. However, the outcome obtained using this method is by no means complete because it contains errors only from the model estimation step. Therefore, post dimension reduction inference is an important topic because it is essential to consider errors from sufficient dimension reduction. In this paper, we review the fundamentals of sufficient dimension reduction methods. Then, we introduce an intuitive and heuristic approach for the recently developed post dimension reduction statistical inference.