By Alan L. T. Paterson
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Extra resources for Amenability
3. only if for every ° = to < tl<···
F(X l ), ... f. d. f. f. corresponding to If m Xl" ",Xn and U n [0,1] is F(Xl ), ... ,F(Xn ) , it follows that UoF n is a statistical functional, then we can define a functional by T(F ) n and T(F) . f. G on [0,1] , we can define L(G) T(GoF) 27 when T(GoF) tional T is defined. Therefore for fixed induces a functional [0,1] concentrated on Let [0,1] e[O,l] e[O,l] [0,1] and and view them as elements of the func- D[O,l] , which we shall now consider in detail. f. f. 's concentrated on tion spaces F, the statistical func- D[O,l] [0,1] [0,1] I G(x) I , G E e[O,l] .
F. f. 10) induces defined by The functional 1: is defined for G near the uniform U € D(O,lJ , and we shall show that under appropriate conditions it is Hadamard differentiable at U. Since L-estimators are explicitly defined, no implicit function theorem will be needed, and Hadamard differentiability can be proved directly. 3 R-estimators R-estimators, or rm1k-estimators, are implicitly defined statistical functionals based on rank statistics. They were introduced by Hodges and Lehmann (1963) and are used to obtain estimates of location in one sample problems and estimates of shift in two sample problems.
Amenability by Alan L. T. Paterson