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By Dynkin E.B.

The Annals of Probability1993, Vol. 21, No. three, pp 1185-1262

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B)1 < 00 doesn't depend on the choice of adjustment function a(·). , ( 1) for any fixed B1 E 8 1 , specifically B1 = Bo. Consistency of M-Estimators and One-Sided Bracketing 39 It follows from the definitions that for an adjustable function p(x, B) and an adjustment function a(x) for it, a(·) and p(·,B) for any B E 8 1 take finite real values for P-almost all x, while for any B E 8, p(x, B) > -00 for P-almost all x. ) is adjustable. In case (II), for the three examples of functions p mentioned, the adjustment function a(·) can be chosen not depending on P.

Almost uniformly) and the value and double consistency hold. 0 Huber (1967, Sec. 2; 1981, Sec. 2), gives a proof of consistency of approximate M-estimators under his conditions (A-I) through (A-5) on functions p(x,B), a(x) and b(B), where b(·) is a positive, continuous function on e. Huber's conditions, and a proof of the following Theorem, are given in Sec. 7 below. 4 Theorem. ), P and Bo, then p E Hs(P, Bo). 3 that under Huber's conditions, as he showed, weak (resp. strong) approximate M-estimators converge in outer probability (resp.

Sequences of capacities, with connections to large deviation theory. J. Theor. , 9:19-35. [4] Shao, Q. M. (1997). Self-normalized large deviations. Ann. , 25:285-328. edu Progress in Probability, Vol. 43 © 1998 Birkhiiuser Verlag Basel/Switzerland Consistency of M-Estimators and One-Sided Bracketing RICHARD M. DUDLEY* ABSTRACT. Some facts in empirical process theory are based on brackets of functions, defined by [I, h] := {g: / ~ 9 ~ h}. For minimization problems arising in M-estimation it is shown that one can use one-sided brackets [/,00).

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