Confidence Bands for Distribution Functions: A New Look at the Law of the Iterated Logarithm

We present a general law of the iterated logarithm for stochastic processes on the open unit interval having subexponential tails in a locally uniform fashion. It applies to standard Brownian bridge but also to suitably standardized empirical distribution functions. This leads to new goodness-of-fit tests and confidence bands which refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter procedures in the tail regions of distributions are essentially preserved while gaining considerably in the central region.