On minimum distance estimation using Kolmogorov-Lévy type metrics

Let X ₁, . . . , X _n be independent identically distributed random variables with a common continuous (cumulative) distribution function (d.f.) F, and F̂ _n the empirical d.f. (e.d.f.) based on X ₁, . . . , X _n. Let G be a smooth d.f. and G _θ = G(· - θ) its translation through θ ∈ ℝ. Using a Kolmogorov-Lévy type metric ρ _α defined on the space of d.f.s on ℝ, the paper derives both null and non-null limiting distributions of √n[ρ _α (F̂ _n, G _θn)-ρ _α(F, G _θ̄)], √n(θ _n - θ̄) and √nρ _α(G _θ, G _θ̄), where θ _n and θ̄ are the minimum ρ _α-distance parameters for F̂ _n and F from G, respectively. These distributions are known explicitly in important particular cases; with some complementary Monte Carlo simulations, they help us clarify our understanding of estimation using minimum distance methods and supremum type metrics. We advocate use of the minimum distance method with supremum type metrics in cases of non-null models. The resulting functionals are Hadamard differentiable and efficient. For small scale parameters the minimum distance functionals are close to medians of the parent distributions. The optimal small scale models result in minimum distance estimators having asymptotic variances very competitive and comparable with best known robust estimators.