6.2 Let F be the family of densities

where f0 is a fixed density (but generally unknown) such that

and f0 has a bounded derivative in a neighborhood of 0.

Consider the linear regression model Yi = x0 i + ei, i = 1, . . . , n with 2 Rp, where e1, . . . , en are independent errors with joint density f 2 F, but unknown. The solution Tn() of the minimization

where (x) = (1 − )x2 + |x|, x 2 R and ˆsn is an estimator of s, is a Mestimator that can be considered as a mixture of the LSE and L1 estimators of . For each f 2 F, there exists an optimal value f 2 [0, 1], leading to the minimal asymptotic variance of Tn(), 0 1. The optimal f can be estimated consistently by an estimator that depends only on ˆsn and on two first sample moments (Dodge and Jurecková (2000)).
As estimators of s = 1 f(0) , we can use either (4.74) or (4.75).

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F = {: f(x) = (), 8 >0}