By Manfred Denker

Ebook through Denker, Manfred

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**Extra resources for Asymptotic Distribution Theory in Nonparametric Statistics**

**Example text**

1 mZ Z ~ and if all ) - ~ i 3) j < . . < km 1 :;: k ' ) - 3) m2 < ... < i :;: n and m :;: n l l , j j Z L n ' If all i are diff erent fr om the Z u jv are different from all l v' 's, then the e x pec t a t i o n und er th e sum i s ze ro . In all other ca s es 2, 21/hIl usi n g Hölder's this e x p e c t a t i o n c an be bounded by i neq u a l i t y . Th e number of these c as e s i s e xac t ly = t 1 ) 2 (n 2 ) 2 ITl, (1 _ ITl 2 (n,-JT1 1 ) · · (n l -2ITl1 + ' ) (n 2- JT1 2)·· (n 2-2 r12 + ' ) ) n, .

1Ob) For e ach 3 t h ree de r i v a t ive s 0 E a f (x, 3) e , the abs olute v a l ue s of the a nd a3 a r e unifo r mly bounded i n som e ne i ghbou rhoo d of 3 0 by functions u (x ) , v (x ) a n d w (x ) r espe ct i ve l y s atis f ying f u (x) dx (2. 1Od) for e very 3 o < o E o < 0 Jv (x ) < co , ( S\ dx and <: ca , I H o g f(x, 3) d3 5=5 )2 o 5 w( x) dF( x) < dF(x,5 0 ) < 00 • = e s E 8. ( df ( x,5) a5 I 3=3 ) 2 (f(x ,3 0 • » -1 dx < = o The max imum li kelihood esti mator of 3 i s d ef ined by the s ol ut i on o f 56 n Hog f ( Xi ,5) a5 I i=l = 0 for whi ch o.

4: A fi eld u, v of rea l- valued random v a ri ab le s u, v (u,v