1 (3.5.1)

3.5 Consistency of approximate M-estimators of ψ type. As in Sec. 3.3, let (X, A, P) be a probability space and Θ a locally compact separable metric space. Let ψ(θ, x) be a function of x in X and θ ∈ Θ with values in a Euclidean space R m. Let X 1 , X 2 ,. .. be independent with values in X and distribution P. A sequence of estimators T n := T n (X 1 ,. .. , X n) with values in Θ will be called approximate M-estimators of ψ type if i=1 If ψ is jointly measurable, as will follow from assumptions to be given, then since estimators T n by definition are assumed to be statistics (measurable functions of the observations), the almost uniform convergence in (3.5.1) will be equivalent to almost sure convergence. Recall that if T n are M-estimators of ψ type, the expression on the left in (3.5.1) equals 0, at least with probabilities converging to 1. Convergence of T n to some θ 0 will be proved under some assumptions as follows. (B-1) For each θ ∈ Θ, the function ψ(θ, ·) is A-measurable. (B-2) For almost all x, ψ(·, x) is continuous on Θ. (B-3) λ(θ) := Eψ(θ, ·) is defined and finite for all θ, and for some θ 0 , λ(θ 0) = 0, while = 0 for all θ λ(θ) = θ 0. (B-4) There is a continuous, positive function b(·) on Θ, bounded away from 0, so that for some b 0 > 0, b(θ) ≥ b 0 for all θ, and (i) Ψ(x) := sup θ |ψ(θ, x)|/b(θ) is integrable, (ii) lim inf θ→∞ |λ(θ)|/b(θ) ≥ 1, and (iii) E{lim sup θ→∞ |ψ(θ, x) − λ(θ)|/b(θ)} < 1. A first question about the assumptions is: how are we to verify them, given that the true distribution P of the observations is unknown? (B-1) and (B-2) don't depend on P , so they can be checked. In (B-3), ψ(θ, ·) will be integrable for all P and θ if it is a bounded function of x for each θ. If ψ is bounded uniformly in x and θ, as for the classes of ψ functions with −A ≤ ψ(θ, x) ≤ A < +∞ considered in the 1-dimensional location case, so much the better. To verify that there is unique θ 0 with λ(θ 0) …