A third-order optimum property of the maximum likelihood estimator

J. Pfanzagl and W. Wefelmeyer

Journal of Multivariate Analysis, 1978, vol. 8, issue 1, 1-29

Abstract: Let [theta](n) denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators [theta](n) + n-1 q([theta](n)), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T(n) there exists q such that the risk of [theta](n) + n-1 q([theta](n)) exceeds the risk of T(n) by an amount of order o(n-1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If q* is chosen such that [theta](n) + n-1 q*([theta](n)) is unbiased up to o(n-1/2), then this estimator minimizes the risk up to an amount of order o(n-1) in the class of all estimators which are unbiased up to o(n-1/2). The results are obtained under the assumption that T(n) admits a stochastic expansion, and that either the distributions have--roughly speaking--densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.

Keywords: Asymptotic; theory; Edgeworth-expansions; higher-order; efficiency; complete; classes; maximum; likelihood; estimation; unbiasedness (search for similar items in EconPapers)
Date: 1978
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Citations: View citations in EconPapers (35)

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