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Example text

0 for all J J=l j = 1, ... , J l 0 there exists (2 > 0 such that for all t E To maxyEm IGt(y) - Fe , t(y)1 ~ implies 2 (2 . 1 provides for all Qe with dK(Qe, PO,a) supp(6) = To S <2 and supp(O C maxOE0 11I(Qe,£i) - lI(Po,a, 0')1 < """ max{3E1W Vl(Qe,t,a(t),{3) ~tETo <1 = <1.

P. )P•• on the Borel O'-field of JR, where 0. E [0,1], P. is a discrete measure and P•• has a density with respect to the Lebesgue measure A. Moreover let denote p. the set of all probability measures on JR x T. Note that if the support supp{ 15) of 15 is finite, which is usually the case in designed experiments, then for every choice Pt E P for t E supp{ 15) there exists a probability measure P® 15 E p. with a Markov kernel P such that Pt = p(" t) for t E supp(c5). Usually the neighbourhoods around P8,6 are regarded but sometimes it will be useful to have neighbourhoods also around other probability measures P6 E p.

Proof. In Huber (1981), p. 2), where 1/;e : IR x T --+ IRq is bounded and continuous and satisfies f 1/;e(y, t)P8Ady, dt) = O. The additional assumption of «P8,d = (0) for all with supp(e) = supp(o) provides in particular for o(c) e (l-c)o+ce o _ lim I«P8,6(f) __ 0 - «Pe,6) - (P8 ,6(f) - P8,6)1 dK(P8,6(f)' P8,6) lim'} __ 0 maxtEsupp(6) 1(1 - c) o( {t}) + c e( {t}) - o( {t})1 (I-C)O({t})j1/;8(y,t)P8t(dY) I'" ~tEsupp(6) , + L:tEsUPP(6) fe({t}) j 1/;e(Y, t) Pe,t(dy)1 I ~tEsuPP(6) e( {t}) maxtEsupp(6) for all e with supp(e) t E supp(o).

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