Nonparametric Likelihood Confidence Bands for a Distribution Function

Abstract

Berk and Jones described a nonparametric likelihood test of uniformity with greater asymptotic Bahadur efficiency than any weighted Kolmogorov–Smirnov test at any alternative to U[0, 1]. We invert this test to form confidence bands for a distribution function using Noé's recursion. Nonparametric likelihood bands are narrower in the tails and wider in the center than Kolmogorov-Smirnov bands and are asymmetric about the empirical cumulative distribution function.

This article describes how to convert a confidence level into a likelihood threshold and how to use the threshold to compute bands. Simple, computation-saving approximations 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 illustrated on some galaxy velocity data and are shown to improve power over Kolmogorov-Smirnov bands on some examples with n = 20.

Key Words:

• Anderson-Darling
• Bahadur efficiency
• Empirical distribution
• Kolmogorov-Smirnov
• Noé's recursion
• Relative optimality