Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 2 , pp. 265--293

THE OPTIMAL LOWER BOUND FOR THE TAIL PROBABILITY OF THE KOLMOGOROV DISTANCE IN THE WORST DIRECTION

By

LI-XING ZHU and PING CHENG, The Chinese Academy of Sciences

SUMMARY. Let Dn: =supF $\sqrt{n}$|Pnf-Pf|, where Pn is an empirical measure based on a d-dimensional iid vectors with elliptical probability measure P, F is the class consisting of all indicator functions of half spaces in d-dimensional Euclidean space. For the supremum of the Gaussian procss, sup|Wp(f)|, indexed by F, which is the weak limit of $\sqrt{n}D_n$, it is shown that for an appropriate c>0 and any l>0, cl(2d-1)e-2l2 P(supF | Wp(f) | >l), d 2. And then for every H>0 and appropriate C(H), cl(2d-1)e-2l2 (1+C(H)(log n)-H) P{Dn l}.

AMS (1992) subject classification. Primary 60E99, 62F20; secondary 60E17.

Key words and phrases. Gaussian process, optimal lower bound, projection pursuit, Kolmogorov distance.

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