By Manfred Denker

Ebook via Denker, Manfred

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Additional info for Asymptotic Distribution Theory in Nonparametric Statistics

Example text

6) lim n"oo p( Ö,E > 0 n P / 2 - ö sup O:>s:o 1 and all P I ddt T(F(t» n P p = 1, . ,m+1 I t=s I> E ) 0 F(t) = tF + (l-t) F. n n ~ k :0 m is the smallest integer such that Suppose that where k ~T(F(t»1 dt k n t=O does not vanish . 3) tend to k/ 2 zero in probability, even after being blown up by n However, the properties of a von Mises' functional are too strong to be satisfied for many statistics. , dkkT(F(t))1 dt n t=O k/2 In many examples we shall show that n Rem (T,F) ·~O k (as n --+ (0) in probabil i ty.

The theorem follows. 5: Let h be a kernel of degree m with 1';: 111 lim sup (n log log n)-1/2 n ( U (h)-5) n->= n < =. s .. 1. 3. 3. 2: Le t h be a ker ne l o f de g r e e m sat isfyi n g (1. 8) max{E lh (X. , ... , X. )1 l1 lm Then ( 1. 3. s. a nd i n More o ver, (1. s. --+ 1/ 2(kU k(h)-k Vk(h))-- 0 =. 1 1) U (h ) n v n (h ) = U (h ) (1 _ m ! (n ) ) - n- m n nrn rn I: h (X. , .. , X . e th e summa t i on e xtends o ver all c hoi c es of incices ' :> i" . Sin c e n -1 1 - m! 2 vn U (h) (l-m! s.

1) n 1/2 Rem (T,F)~O 1 imply that in probability. Two results are given here. 1: T: be a functional defined V~R V c Cb(R), the space of bounded real valued on an open subset functions on R equipped wi th the s upremum - norm is Fr~chet n 1/2 Rem (T, F) ---+ 0 (as n 1 in probabili ty. f. 8) 2 - T (F) - T (G- F) I 11 G-Fil oo IIFn - FII00 0(1) on an open subset = o. 2 : I ItFn - Fi the Kolmogorov-Smirnov statistic Since n 1/ I T (G) lim 11 G-FII ~o 00 V c G = F T: V ----tR 0«(0,1]) n. be a functional defined where 0«(0,1]) is considered as the complete, non-separable Banach space with the supremum - norm.