Abstract
We introduce new goodness-of-fit tests and corresponding confidence bands for distribution functions. They are inspired by multiscale methods of testing and based on refined laws of the iterated logarithm for the normalized uniform empirical process and its natural limiting process, the normalized Brownian bridge process . 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 ϕ-divergences.
Funding Statement
The first author was supported in part by the Swiss National Science Foundation. The second author was supported in part by NSF Grant DMS-1104832 and NI-AID Grant 2R01 AI291968-04.
Acknowledgments
The authors owe thanks to David Mason for pointing out the relevance of the tools of Csörgő et al. [5] for some of the results presented here. We are also grateful to Günther Walther for stimulating conversations about likelihood ratio tests in nonparametric settings and to Rudy Beran for pointing out the interesting results of Bahadur and Savage [2]. Constructive comments of two referees and an associate editor are gratefully acknowledged.
Citation
Lutz Dümbgen. Jon A. Wellner. "A new approach to tests and confidence bands for distribution functions." Ann. Statist. 51 (1) 260 - 289, February 2023. https://doi.org/10.1214/22-AOS2249
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