  Quantiles can represent key operational and business metrics, but the
computational challenges associated with inference has hampered their adoption
in online experimentation. One-sample confidence intervals are trivial to
construct; however, two-sample inference has traditionally required
bootstrapping or a density estimator. This paper presents a new two-sample
difference-in-quantile hypothesis test and confidence interval based on a
likelihood-ratio test statistic. A conservative version of the test does not
involve a density estimator; a second version of the test, which uses a density
estimator, yields confidence intervals very close to the nominal coverage
level. It can be computed using only four order statistics from each sample.
