Distribution-Free Conditional Median Inference
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We consider the problem of constructing confidence intervals for the median of a response Y ∈ℝ conditional on features X = x ∈ℝ^d in a situation where we are not willing to make any assumption whatsoever on the underlying distribution of the data (X,Y). 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. Further, we provide 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.
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