On post dimension reduction statistical inference

Kyongwon Kim, Bing Li, Zhou Yu, Lexin Li

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


The methodologies of sufficient dimension reduction have undergone extensive developments in the past three decades. However, there has been a lack of systematic and rigorous development of post dimension reduction inference, which has seriously hindered its applications. The current common practice is to treat the estimated sufficient predictors as the true predictors and use them as the starting point of the downstream statistical inference. However, this naive inference approach would grossly overestimate the confidence level of an interval, or the power of a test, leading to the distorted results. In this paper, we develop a general and comprehensive framework of post dimension reduction inference, which can accommodate any dimension reduction method and model building method, as long as their corresponding influence functions are available. Within this general framework, we derive the influence functions and present the explicit post reduction formulas for the combinations of numerous dimension reduction and model building methods. We then develop post reduction inference methods for both confidence interval and hypothesis testing. We investigate the finite-sample performance of our procedures by simulations and a real data analysis.

Original languageEnglish
Pages (from-to)1567-1592
Number of pages26
JournalAnnals of Statistics
Issue number3
StatePublished - Jun 2020

Bibliographical note

Funding Information:
Acknowledgments. We thank the Editor, the Associate Editor and two referees for their prompt and thoughtful reviews, which contain many useful comments and suggestions that have helped us to improve this paper. The second author was supported in part by the NSF Grant DMS-1713078. The third author was supported in part by the National Natural Science Foundation of China 11831008, 11571111. The fourth author was supported in part by the NSF Grant DMS-1613137 and NIH Grant AG034570.

Publisher Copyright:
© Institute of Mathematical Statistics, 2020.


  • Central subspace
  • Directional regression
  • Estimating equations
  • Generalized method of moment
  • Influence function
  • Sliced inverse regression
  • Von mises expansion


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