Melded Confidence Intervals Do Not Provide Guaranteed Coverage

Melded confidence intervals were proposed as a way to combine two independent one-sample confidence intervals to obtain a two-sample confidence interval for a quantity like a difference or a ratio. Simulation-based work has suggested that melded confidence intervals always provide at least the nominal coverage. However, we show here that for the case of melded confidence intervals for a difference in population quantiles, the confidence intervals do not guarantee the nominal coverage. We derive a lower bound on the coverage for a one-sided confidence interval, and we show that there are pairs of distributions that make the coverage arbitrarily close to this lower bound. One specific example of our results is that the 95% melded upper bound on the difference between two population medians offers a guaranteed coverage of only 88.3% when both samples are of size 20.