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An identity for multidimensional continuous exponential families and its applications

Author

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  • Chou, Jine-Phone

Abstract

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.

Suggested Citation

  • Chou, Jine-Phone, 1988. "An identity for multidimensional continuous exponential families and its applications," Journal of Multivariate Analysis, Elsevier, vol. 24(1), pages 129-142, January.
  • Handle: RePEc:eee:jmvana:v:24:y:1988:i:1:p:129-142
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    Cited by:

    1. Pommeret Denys, 2002. "Information Inequalities For The Risks In Simple Quadratic Exponential Families," Statistics & Risk Modeling, De Gruyter, vol. 20(1-4), pages 81-94, April.
    2. Papadatos, N. & Papathanasiou, V., 1998. "Variational Inequalities for Arbitrary Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 154-168, November.
    3. Nair, N. Unnikrishnan & Sankaran, P.G., 2008. "Characterizations of multivariate life distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 2096-2107, October.
    4. Hornik, Kurt & Grün, Bettina, 2014. "On standard conjugate families for natural exponential families with bounded natural parameter space," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 14-24.

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