NONPARAMETRIC LIKELIHOOD CONFIDENCE BANDS FOR A DISTRIBUTION FUNCTION

Berk and Jones described a nonparametric likelihood test of uniformity with greater asymptotic Bahadur efficiency than any weighted Kolmogor ov-Smirnov test at any alternative to U[0, 1]. We invert this test to form confidence bands for a distribution function using Noe's recursio n. Nonparametric likelihood bands are narrower in the tails and wider in the center than Kolmogorov-Smirnov bands and are asymmetric about t he empirical cumulative distribution function. This article describes how to conv ert a confidence lev el into a likelihood threshold and ho w to use the threshold to compute bands. Simple. computation-saving ap proximations to the threshold are given for confidence levels 95% and 99% and all sample sizes up to 1,000. These yield coverage between the nominal and .01% over the nominal. The likelihood bands are illustrat ed on some galaxy velocity data and are shown to improve power over Ko lmogorov-Smirnov bands on some examples with n = 20.