A linear mini-max estimator for the case of a quartic loss function (Q2722267)

The following linear estimation problem is considered: Let \(Y(t)\) be a stochastic process on the interval \([0,1]\), given by NEWLINE\[NEWLINEdY(t)=\theta(t)dt+dW(t),NEWLINE\]NEWLINE where \(W\) is a standard Wiener process and \(\theta\in\Theta\subset L_{2}[0,1]\) is an unknown function. A bounded continuous linear function \(L\) on \(\Theta\) is specified, and the statistical problem is to estimate \(L(\theta)\) by a linear estimator \(\langle m,Y\rangle=\int_{0}^{1}m(t)dY(t)\) for the case of a quartic loss function. The problem for a quadratic loss function was first addressed by \textit{A.D. Ibragimov} and \textit{R.Z. Khas'minskij} [Teor. Veroyatn. Primen. 29, No.1, 19-32 (1984; Zbl 0532.62061)] who obtained a minimax theorem as well as a linear minimax estimator in this case. In Lect. Notes Econ. Math. Syst. 389, 9-23 (1992; Zbl 0792.62075), the authors used convex analysis to analyze the quadratic case. The basic idea of their proof for the quadratic case carries over to the case of a quartic loss function. Following the idea of Ibragimov and Khas'minski, a lower bound for the ratio of the minimax risks is derived for all nonlinear estimators versus all linear estimators for the quartic case.