Loss functions for loss estimation (Q1120205)

Let X be a random variable with distribution \(P_{\theta}\), where \(\theta\in \Theta\), and d(X) a decision for \(\theta\) with loss W(\(\theta\),d(X)). A class of loss functions combining the decision error W(\(\theta\),d(X)) and the error in estimating W(\(\theta\),d(X)) by h(X) is introduced. Under these loss functions the Bayes procedure \((d_ B(X),h_ B(X))\) has the following form: \(d_ B(X)\) is just the Bayes decision corresponding to the loss W(\(\theta\),d(X)) and \(h_ B(X)=E(W(\theta,d_ B(X))| X)\). Under some regularity assumptions, a necessary and sufficient condition for (d(X),h(X)) to be admissible is given.