By Peter J. Huber
Here's a short, and easy-to-follow creation and assessment of strong information. Peter Huber focuses totally on the $64000 and obviously understood case of distribution robustness, the place the form of the real underlying distribution deviates a little from the assumed version (usually the Gaussian law). an extra bankruptcy on fresh advancements in robustness has been additional and the reference checklist has been increased and up-to-date from the 1977 variation.
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Fo . Then = 1 A (Fo, 1/Jo) 8. Let F, = (1 - t)Fo + tFt, I(Ft ) < oo. Then, explicit calculation gives [d 1 dt A (F,, 1/10)J - r=o = = l(Fo). [ d I(F,) dt J - t=O 2:: 0. - It follows that the asymptotic variance A (F, 1/10) attains its maximum over PJ at F0, but there the M-estimate is asymptotically efficient. -estimate for location based on F0 is minimax over P/J. Example. /2; with l x l � c. where c = c (e ). The L- and R-estimates which are efficient at F0 do not necessarily yield minimax solutions, since convexity fails (point 6 in the above sketch of the proof).
Note that (Xw(i))1;:;; ; ;:;; n is a Winsorized sample. 4) where iw (1/n ) L Xw(i) is the Winsorized mean. 4) for the variance estimate remains valid, if we define x8 +1 by linearly interpolating between the two adjacent order statistics x lg+IJ and x fg+Il, and similarly for Xn -g· If g is an integer, then � T" is the trimmed mean, but otherwise � is a kind of Winsorized mean. 4) coincides with J IC(x, F", i.. l dF". 3). = = This page intentionally left blank CHAPTER IV Asymptotic Minimax Theory 11.
L/l(x ) X FIG . 4 The values o f c and b o f course depend on e . The actual performance does not depend very much o n exactly how 1/1 rede scends to 0, only one should make sure that it does not do it too steeply; in particular 1 1/1'1 should be small when / 1/1 / is large. e. (X;, Tn) = 0. e. (x, 6)/(x, 6) dx = 0. (x, 8ff(x, 8 ) dx A (F6, "' )= (J 1/! (x, 8 )(aj a8)f(x, O) dxi · Hampel's extremal problem now is to put a bound on the gross error sensitivity: sup IIC(x; FIJ, nl � kll X with some appropriately chosen function kll and, subject to this side condition, to minimize the asymptotic variance A (F6, 1/1 ) at the model.