Ghosh, J. K. ; Sinha, B. K. ; Wieand, H. S. (1980) Second order efficiency of the MLE with respect to any bounded bowlshape loss function Annals of Statistics, 8 (3). pp. 506521. ISSN 00905364

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Abstract
Let X_{1}, X_{2}, .. be a sequence of i.i.d. random variables, each having density f(x, θ_{0}) 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 θ_{0}. Then (under certain regularity conditions) one can choose a function c(θ) such that θˆ = θˆ + c(θˆ)/n satisfies Pθ_{0} {x_{1}< n^{½}(θˆ'  θ_{0})(I(θ_{0}))^{½} < x_{2}} > Pθ_{0} {x_{1}< n^{½}(T  θ_{0})(I(θ_{0}))^{½} < x_{2}} + o(n^{1}), for all x_{1}, x_{2} > 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 bowlshaped 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).
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