Convergence rates for empirical Bayes two-action problems: The uniform \(U(0,{\theta{}})\) distribution
This paper deals with the convergence rate of a sequence of empirical Bayes decision rules when the observations are uniformly distributed over \((0,\theta)\), where \(\theta\) is a random variable having an unknown prior distribution on \((0,b\leq\infty)\). The goal is to choose between the hypothesis \(H_ 0\): \(\theta\leq\theta_ 0\) and \(H_ 1\): \(\theta>\theta_ 0\), where \(\theta_ 0\) is given such that \(0<\theta_ 0<b\). The loss function is \(L_ 0(\theta)=(\theta-\theta_ 0)^ +\) if \(H_ 0\) is chosen and \(L_ 1(\theta)=(\theta_ 0-\theta)^ +\) if \(H_ 1\) is chosen, where \((U)^ +=\max(U,0)\). Let \(x_ 1,\dots,x_ n\) be the previous observations and let \(x_{n+1}=x\) denote the current observation. The proposed empirical Bayes rule is to select \(H_ 0(H_ 1)\) if \(\delta_ n(x)\leq(>)0\), where \(\delta_ n(x)=1-F_ n(x)+(x-\theta_ 0)f_ n(x)\). Here \[ F_ n(x)=n^{-1}\sum_ 1^ n I(x_ i\leq x), \qquad f_ n(x)=(F_ n(x+h_ n)-F_ n(x))/h_ n \] for \(n\geq 1\) and \(\{h_ n\}\) is a sequence of positive real numbers such that \(h_ n\to 0\) and \(nh_ n\to\infty\), as \(n\to\infty\). It is shown that the given rule is asymptotically optimal and that the order of the associated convergence rate is \(O(n^{-\alpha})\) for some constant \(\alpha\) such that \(0<\alpha<1\).
scientific article; zbMATH DE number 94156