Let X1,... ,Xn be a random sample from the distribution with distribution function F(n)(x − θ), where
F(n) ≡ (1 − cn−1/2 )G + cn−1/2H,
G and H are fixed distribution functions, G(x)+G(−x) = 1, ∀x ∈ R1. Let Mn be an M-estimator defined as a solution of Pn i=1 ψ(Xi−M) = 0, where ψ is a nondecreasing odd function such that ψ ′ and ψ ′′ are continuous and bounded up to a finite number of points. Then
n 1 2 (Mn − θ) D→ N (b,ν2 (ψ,G)),
where b = cA/IEGψ ′ , A = R ψ(x)dH(x), and ν 2 (ψ,G) is defined in (3.20). hence, Mn has asymptotic bias b and the asymptotic mean square error ν 2 (ψ,G) + b 2 .