Minimax quadratic estimation of a quadratic functional
Let \(f\) be a real function and consider its noisy sample observations \(v_ i=f(t_ i)+z_ i\), \(i=1,2,...,n\), where \(t_ i=-\pi +2\pi (i/n)\) are equispaced on \([-\pi,\pi]\) and the noise terms \(z_ i\) are iid \(N(0,\sigma^ 2)\). Let \(f^{(k)}(t)\) be the \(k\)th derivative of \(f\). The authors consider the estimation of the quadratic functional \(Q(f)=\int^{\pi}_{-\pi}(f^{(k)}(t))^ 2\,dt\) and derive a simple quadratic estimate which is asymptotically minimax among quadratic estimates. The given estimate is also shown to be rate optimal.
asymptotically minimax among quadratic estimates
scientific article; zbMATH DE number 4194942
scientific article