An identity for multidimensional continuous exponential families and its applications

Let Z be a p-dimensional continuous random vector with density f[mu](z) = e[mu]·z - M([mu]) - K(z)1E(z). Subject to some conditions on f[mu] and g the identity E[mu]([backward difference]K(Z) - [mu]) g(Z) = E[mu][backward difference]g(Z) holds. Through use of the identity classes of estimators are found to improve on the unbiased estimator [backward difference]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) E{E(X[theta]) X = x} = ax + b; (ii) E{(E(X[theta]) - (ax + b))(E(X[theta]) - (ax + b))' X = x} = cE{[backward difference]E(X[theta]) X = x}; and if X has a quadratic variance function (iii) E{E(X[theta])n X = x} = Pn(x) an nth-degree polynomial of x.