Ab. Owen, NONPARAMETRIC LIKELIHOOD CONFIDENCE BANDS FOR A DISTRIBUTION FUNCTION, Journal of the American Statistical Association, 90(430), 1995, pp. 516-521
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.