We study the problem of loss estimation that involves for an observable \(X \sim f_{\theta }\) the choice of a first-stage estimator \(\hat{\gamma }\) of \(\gamma (\theta )\) , incurred loss \(L=L(\theta , \hat{\gamma })\) , and the choice of a second-stage estimator \(\hat{L}\) of L. We consider both: (i) a sequential version where the first-stage estimate and loss are fixed and optimization is performed at the second-stage level, and (ii) a simultaneous version with a Rukhin-type loss function designed for the evaluation of \((\hat{\gamma }, \hat{L})\) as an estimator of \((\gamma , L)\) . We explore various Bayesian solutions and provide minimax estimators for both situations (i) and (ii). The analysis is carried out for several probability models, including multivariate normal models \(N_d(\theta , \sigma ^2 I_d)\) with both known and unknown \(\sigma ^2\) , Gamma, univariate and multivariate Poisson, and negative binomial models, and relates to different choices of the first-stage and second-stage losses. The minimax findings are achieved by identifying a least favourable sequence of priors and depend critically on particular Bayesian solution properties, namely situations where the second-stage estimator \(\hat{L}(x)\) is constant as a function of x.
