Mathematics – Statistics Theory
Scientific paper
2006-03-10
Annals of Statistics 2007, Vol. 35, No. 5, 2018-2053
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/0009053607000000244 the Annals of Statistics (http://www.imstat.org/aos/) by the Ins
Scientific paper
10.1214/0009053607000000244
A unified family of goodness-of-fit tests based on $\phi$-divergences is introduced and studied. The new family of test statistics $S_n(s)$ includes both the supremum version of the Anderson--Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47--59] as special cases ($s=2$ and $s=1$, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47--59] and Wellner and Koltchinskii [High Dimensional Probability III (2003) 321--332. Birkh\"{a}user, Basel] for the Berk--Jones statistic applies to the whole family of statistics $S_n(s)$ with $s\in[-1,2]$. On the side of power behavior, we study the test statistics under fixed alternatives and give extensions of the ``Poisson boundary'' phenomena noted by Berk and Jones for their statistic. We also extend the results of Donoho and Jin [Ann. Statist. 32 (2004) 962--994] by showing that all our new tests for $s\in[-1,2]$ have the same ``optimal detection boundary'' for normal shift mixture alternatives as Tukey's ``higher-criticism'' statistic and the Berk--Jones statistic.
Jager Leah
Wellner Jon A.
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