INTRINSIC LOSSES

Since the choice of a particular loss function strongly influences the resulting inference, it seems necessary to rely on ''intrinsic'' loss es when no information is available about the utility function of the decision-maker, rather than to call for classical losses like the squa red error loss. Since this setting is quite similar to the derivation of noninformative priors in Bayesian analysis, we first recall the con ditions of this derivation and deduce from these conditions some requi rements on the intrinsic losses. It then appears that these loss funct ions should only depend on the sampling distribution and that they sho uld be independent of the parameterization of the distribution. The re sulting estimators are therefore transformation equivariant. We study the properties of two natural intrinsic losses, namely entropy and Hel linger losses, and show that they can be expressed in closed form for exponential families. Moreover, the entropy loss also provides analyti c expressions of Bayes estimators under conjugate priors; the derivati on of Bayes estimators associated with the Hellinger loss is more cumb ersome, as shown in Poisson and Gamma cases, while leading to similar estimators.