CHARACTERISATION OF LINEAR MINI-MAX ESTIMATORS FOR LOSS FUNCTIONS OF ARBITRARY POWER

K. Helmes () and C. Srinivasan ()
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K. Helmes: Institut für Operations Research, Humboldt — Universität zu Berlin, Spandauer Straße 1, 10178 Berlin, Germany
C. Srinivasan: The Department of Statistics, College of Arts and Suences, University of Kentucky, Lexington, Kentucky, 40506-0027, USA

International Game Theory Review (IGTR), 2001, vol. 03, issue 02n03, 203-211

Abstract: LetY(t),t∈[0,1], be a stochastic process modelled asdYt=θ(t)dt+dW(t), whereW(t)denotes a standard Wiener process, andθ(t)is an unknown function assumed to belong to a given setΘ⊂L2[0,1]. We consider the problem of estimating the valueℒ(θ), where ℒ is a continuous linear function defined on Θ, using linear estimators of the form =∫m(t)dY(t),m∈L2[0,1]. The distance between the quantityℒ(θ)and the estimated value is measured by a loss function. In this paper, we consider the loss function to be an arbitrary even power function. We provide a characterisation of the best linear mini-max estimator for a general power function which implies the characterisation for two special cases which have previously been considered in the literature, viz. the case of a quadratic loss function and the case of a quartic loss function.

JEL-codes: B4 C0 C6 C7 D5 D7 M2 (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1142/S0219198901000397

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