A New Approach to Tests and Confidence Bands for Distribution Functions

Abstract: 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.