Abstract
We consider the problem of constructing confidence intervals for the median of a response conditional on features in a situation where we are not willing to make any assumption whatsoever on the underlying distribution of the data . We propose a method based upon ideas from conformal prediction and establish a theoretical guarantee of coverage while also going over particular distributions where its performance is sharp. Additionally, we prove an equivalence between confidence intervals for the conditional median and confidence intervals for the response variable, resulting in a lower bound on the length of any possible conditional median confidence interval. This lower bound is independent of sample size and holds for all distributions with no point masses.
Funding Statement
D. M. thanks Stanford University for supporting this research as part of the Masters program in Statistics. E. C. was supported by Office of Naval Research grant N00014-20-12157, by the National Science Foundation grant OAC 1934578, and by the Army Research Office (ARO) under grant W911NF-17-1-0304.
Acknowledgments
We thank Lihua Lei for advice on different approaches and recommended resources on this topic, as well as the Stanford Statistics department for listening to a preliminary version of this work. We also thank a referee for encouraging us to deepen the connection between predictive and conditional median confidence intervals.
Citation
Dhruv Medarametla. Emmanuel Candès. "Distribution-free conditional median inference." Electron. J. Statist. 15 (2) 4625 - 4658, 2021. https://doi.org/10.1214/21-EJS1910
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