Second order efficiency of the MLE with respect to any bounded bowl-shape loss function

Abstract

Let X₁, X₂, .. be a sequence of i.i.d. random variables, each having density f(x, θ₀) where {f(x, θ)} is a family of densities with respect to a dominating measure µ. Suppose n^½(θˆ - θ) and n^½(T - θ), where θˆ is the mle and T is any other efficient estimate, have Edgeworth expansions up to o(n^-1) uniformly in a compact neighbourhood of θ₀. Then (under certain regularity conditions) one can choose a function c(θ) such that θˆ = θˆ + c(θˆ)/n satisfies Pθ₀ {-x₁< n^½(θˆ' - θ₀)(I(θ₀))^½ < x₂} > Pθ₀ {-x₁< n^½(T - θ₀)(I(θ₀))^½ < x₂} + o(n^-1), for all x₁, x₂ > 0. This result implies the second order efficiency of the mle with respect to any bounded loss function Ln(θ, a) = h(n^½(a - θ)), which is bowl-shaped i.e., whose minimum value is zero at a - θ = 0 and which increases as |a - θ| increases. This answers a question raised by C. R. Rao (Discussion on Professor Efron's paper).