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Let Y(t), t∈[0,1], be a stochastic process modelled asdYt=θ(t)dt+dW(t), where W(t) denotesa standard Wiener process, and θ(t) is an unknown function assumedto belong to a given set Θ⊂L2[0,1]. Weconsider the problem of estimating the value ℒ(θ), whereℒ is a continuous linear function defined on Θ, using linearestimators of the form =∫m(t)dY(t),m∈L2[0,1]. The distancebetween the quantity ℒ(θ) and the estimated value ismeasured by a loss function. In this paper, we consider the lossfunction to be an arbitrary even power function. We provide acharacterisation of the best linear mini-max estimator for a generalpower function which implies the characterisation for two specialcases which have previously been considered in the literature, viz.the case of a quadratic loss function and the case of a quartic lossfunction.
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