New PDF release: Asymptotic Distribution Theory in Nonparametric Statistics

By Manfred Denker

ISBN-10: 3528089059

ISBN-13: 9783528089054

ISBN-10: 3663142299

ISBN-13: 9783663142294

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Extra info for Asymptotic Distribution Theory in Nonparametric Statistics

Sample text

9) implies that ' there exists a k o such that 1 2 1 T(F ) - T(F) -T' (Fnk-F) I = n k k n / for all 54 k ~ k o In~/2 Rem1 (T,F) I < This contradicts the assumption. E. We shall now discuss a few examples in the remaining pa rt of this section. Some basic facts from calculus are not explicit- ly stated but are obvious in each case . For example, condi- tions for interchanging differentiation and integration, the fact that S f' (x) dx 0 for a differentiable density f etc. We shall not give all computations in each example.

X 1,X 2, 1. n-m+1/2 (n) ~ n-m+e+1/2 ~ n-1/2. 11) holds for eaeh I whieh is not a partition into points. 9). ( 1 . 3 . 9 ) . The following results are immediate eorollaries to the last proposition. 6: ( 1 . 3. 8). 2 and Theorem 1. 3. 9)~ Let h be a kernel of degree m satisfyinq Then n- 1/ 2 (V (h) - 5) ~ N (0,m2~1) n weakly as n ~ "". 9). 8) and ~1 > O. Define Yn(t} = f (nm 2 ~1) ~1/2 1. 10) <00, m i f t = kin by linear interpolation elsewhere and Zn(t} = Yn([ntJ/n) Then Yn and Zn (n weakly to the standard Wiener process.

6). We leave t h i s as an e xercise: Both der i vati ves c a n be written a s a s um of V-statistics with degen e r ate k e r ne l s. Then us e t he fact that for such a sta- t ist ic wi th degen er ate kerne l h of degre e p (cf. 2: n ~ O. Man y differentiable statistical functionals are defin ed implicitly, diti on. 2 -6 V (h) ) 2 for e xample by some minimiz ing con- One of th em is the maximum lik elihood estim ator Let (ML -esti mato r ). 8 eRbe open and let { F(·,3) be a f a mi ly o f d.

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Asymptotic Distribution Theory in Nonparametric Statistics by Manfred Denker

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