Inference for Median and a Generalization of HulC

Constructing distribution-free confidence intervals for the median, a classic problem in statistics, has seen numerous solutions in the literature. While coverage validity has received ample attention, less has been explored about interval width. Our study breaks new ground by investigating the width of these intervals under non-standard assumptions. Surprisingly, we find that properly scaled, the interval width converges to a non-degenerate random variable, unlike traditional intervals. We also adapt our findings for constructing improved confidence intervals for general parameters, enhancing the existing HulC procedure. These advances provide practitioners with more robust tools for data analysis, reducing the need for strict distributional assumptions.