ABSTRACT
For the test functiorls -5" : it C Z l ), appropriate to t,otal variation approximation, the geiicritl 1)olnitls
were p~xmxl in Barhour, Chen, and Loh (1992); I~ecausc of the exponential factor. these arc only useful for snlall X. I-Iowever, lmtlcr the additional conctitio~t or] the p, that
they showed that
H;.\' (X, p ) < inin {l3 & } ; where r q - pl - 2p2 These bo~mcls are rnuc11 better, but that on Ml is st,ill wcalier than coultl be hoped h , becansc of the logaritlirrtic factor. This f x t o r was rernovecl in Barbow ;md Utcv (1998, Theorem 3.1), in the sctt,irrg of approsima,t,ioi with respect to the Koln~ogorov distance dlc clefirlcd by
dic(I', Q ) := s i q ~ 1 P/Y, pm) --- Q[Z, <XI)/, c
by way of intricate Wuricr argurnenis, hut only at t,l~e xpense of worsenilig the constant factors to a point a t wliich they became ui~itttr;tctive for pritctical application.
