An identity for multidimensional continuous exponential families and its applications (Q1097608)

Let Z be a p-dimensional continuous random vector with density \(f_{\mu}(z)=e^{\mu z-M(\mu)-K(z)}1_ E(z)\). Subject to some conditions on \(f_{\mu}\) and g the identity \[ E_{\mu}(\nabla K(Z)- \mu)g(Z)=E_{\mu}\nabla g(Z) \] holds. Through use of the identity classes estimators are found to improve on the unbiased estimator \(\nabla K(Z)\) of \(\mu\) for an arbitrary quadratic loss function. In another direction, let X be a random vector distributed according to an exponential family with natural parameter \(\theta\) ; if \(\theta\) has a conjugate prior the identity gives: \[ (i)\quad E\{E(X| \theta)| X=x\}=ax+b; \] \[ (ii)\quad E\{(E(X| \theta)-(ax+b))(E(X| \theta)-(ax+b))'| X=x\}=cE\{\nabla E(X| \theta)| X=x\}; \] and if X has a quadratic variance function \[ (iii)\quad E\{E(X| \theta)^ n| X=x\}=P_ n(x) \] an n-th-degree polynomial of x.