  We introduce new goodness-of-fit tests and corresponding confidence bands for
distribution functions. They are inspired by multi-scale methods of testing and
based on refined laws of the iterated logarithm for the normalized uniform
empirical process $\mathbb{U}_n (t)/\sqrt{t(1-t)}$ and its natural limiting
process, the normalized Brownian bridge process $\mathbb{U}(t)/\sqrt{t(1-t)}$.
The new tests and confidence bands refine the procedures of Berk and Jones
(1979) and Owen (1995). Roughly speaking, the high power and accuracy of the
latter methods in the tail regions of distributions are essentially preserved
while gaining considerably in the central region. The goodness-of-fit tests
perform well in signal detection problems involving sparsity, as in Ingster
(1997), Donoho and Jin (2004) and Jager and Wellner (2007), but also under
contiguous alternatives. Our analysis of the confidence bands sheds new light
on the influence of the underlying $\phi$-divergences.
