Asymptotic local minimaxity in sequential point estimation
Let \(\{X_ n\), \(n\geq 1\}\) be a sequence of i.i.d. random variables with mean \(\theta\) and finite variance \(\sigma^ 2\), depending on unknown parameter \(\omega\in \Omega\). The author describes a stopping time t for which the risk \[ R_ A(t,\omega)=E_{\omega}\{A\gamma^ 2_ 0(\omega)(\bar X_ t-\theta)^ 2+t\} \] is as small as possible, where \(A>0\), \(\gamma_ 0\) is a positive function on \(\Omega\), and \(\bar X_ t=(X_ 1+...+X_ t)/t\). If \(t=n\) is a fixed sample size, then \(2\sqrt{A}(\gamma_ 0\sigma)\) is a lower bound for \(R_ A(n,\omega)\), \(n\geq 1\), and the regret of a stopping time t is defined to be \[ r_ A(t,\omega)=R_ A(t,\omega)-2\sqrt{A}(\gamma_ 0\sigma). \] The main results determine an asymptotic lower bound, as \(A\to \infty\), for the minimax regret \(M_ A(\Omega_ 0)=\inf_{t}\sup_{\omega \in \Omega_ 0}r_ A(t,\omega)\) for neighbourhoods \(\Omega_ 0\) of arbitrary parameter points \(\omega_ 0\in \Omega\). The bound is obtained for multiparameter exponential families and the nonparametric case.
scientific article; zbMATH DE number 3930175