Optimality of linear algorithms in ``worst correlation'' problem
The indirect measurements of a vector \(q\in R^ m\) are assumed to be given, \(z_ i=h^ T_ iq+\rho_ i\), where \(h_ i\in R^ m\), \(\rho_ i\in R^ 1\) is a random measurement interference. The linear envelope of the vectors \(h_ i\), \(i=1,...,N\) is assumed to coincide with \(R^ m\); otherwise one passes to a correspondent subspace. Further it is assume that \(\rho =(\rho_ 1,...,\rho_ N)^ T\in P\) where \(P=\{\rho | M\rho_ i=0\), \(M\rho^ 2_ i\leq \sigma^ 2_ i\), \(i=1,...,N\}\), \(\sigma_ i>0\) are given numbers. The objective is to construct an estimate of a scalar quantity \(l=a^ T\cdot q\), \(a\in R^ m\) being a given vector. All kind of estimates \(\hat l=S(z_ 1,...,z_ N)\) of the scalar quantity l are considered. The mapping S: \(R^ N\to R^ 1\) will be called an admissible estimator if \(S(z_ 1,...,z_ N)\) is a Borel function. An optimal estimator \(S^*\) such that \[ \sup_{q\in R^ m,\rho \in P}M| \ell -s^ 4(z_ 1,...,z_ N)|^ 2=\inf_{S}\sup_{q\in R^ m,\rho \in P}| \ell -S(z_ 1,...,z_ N)|^ 2, \] where S is an arbitrary admissible estimator, is given.
scientific article; zbMATH DE number 4150055