Rates of Convergence of Minimum Distance Estimators and Kolmogorov's Entropy

Let (X, A) be a space with a σ-field, M = ₛ; s ∈ Θ\ be a family of probability measures on A with Θ arbitrary, X₁, ⋯, Xₙ i.i.d. observations on P_θ. Define μₙ(A) = (1/n) ∑ⁿi = 1 IA(Xᵢ), the empirical measure indexed by A ∈ A. Assume Θ is totally bounded when metrized by the L₁ distance between measures. Robust minimum distance estimators θ̂ₙ are constructed for θ and the resulting rate of convergence is shown naturally to depend on an entropy function for Θ.