ghyt (44) - distribution P =(1)P P with distribution function F =(1)F F we have lim 0 F 1(v F 1(v = P q v = 1 f(F 1(v F(F 1(v v Hence for P =(1)P P(1 2

ghyt (44) - distribution P =(1)P P with distribution...

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distribution P = (1 − )P + P˜, with distribution function F = (1 − )F + F˜, we have lim ↓0 F −1 (v) − F −1 (v) = P q˜ v = − 1 f(F −1(v)) F˜(F −1 (v)) − v . Hence, for P = (1 − )P + P˜, (1 − 2α) lim ↓0 Q((1 − )P + P˜) − Q(P) = (1 − α)P q˜ 1−α − αP q˜ α − Z 1−α α vdP q˜ v = Z 1−α α 1 f(F −1(v)) F˜(F −1 (v)) − v dv = Z F −1 (1−α) F −1(α) 1 f(u) F˜(u) − F(u) dF(u) = Z F −1 (1−α) F −1(α) F˜(u) − F(u) du = (1 − 2α)Pl ˜ P , where lP (x) = − 1 1 − 2α Z F −1 (1−α) F −1(α) l{x ≤ u} − F(u) du. We conclude that, under regularity conditions, the α-trimmed mean is asymptotically linear with the above influence function lP , and hence asymptotically normal with asymptotic variance Pl2 P . 6.5 Asymptotic relative efficiency In this section, we assume that the parameter of interest is real-valued: γ Γ R. Definition Let Tn,1 and Tn,2 be two estimators of γ, that satisfy √ n(Tn,j − γ) Dθ −→N (0, Vθ,j ), j = 1, 2. Then e2:1 := Vθ,1 Vθ,2 is called the asymptotic relative efficiency of Tn,2 with respect to Tn,1. If e2:1 > 1,

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