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Bayes and empirical Bayes estimation with errors in variables

Author

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  • Zhang, Shunpu
  • Karunamuni, Rohana J.

Abstract

Suppose that the random variable X is distributed according to exponential families of distributions, conditional on the parameter [theta]. Assume that the parameter [theta] has a (prior) distribution G. Because of the measurement error, we can only observe Y = X + [var epsilon], where the measurement error [theta] is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of [theta] based on the contaminated observation Y. We obtain an expression for the Bayes estimator when the prior G is known. For the case G is completely unknown, an empirical Bayes estimator is proposed based on a sequence of observations Y1, Y2,...,Yn, where Yi's are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.

Suggested Citation

  • Zhang, Shunpu & Karunamuni, Rohana J., 1997. "Bayes and empirical Bayes estimation with errors in variables," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 23-34, April.
    • Handle: RePEc:eee:stapro:v:33:y:1997:i:1:p:23-34
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    References listed on IDEAS

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    1. R. Singh & Laisheng Wei, 1992. "Empirical Bayes with rates and best rates of convergence in u(x)C(θ) exp(-x/θ)-family: Estimation case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(3), pages 435-449, September.
    2. J.C. van Houwelingen & Th. Stijnen, 1983. "Monotone empirical Bayes estimators for the continuous one‐parameter exponential family," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 37(1), pages 29-43, March.
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    Cited by:

    1. Pensky Marianna & Alotaibi Mohammed, 2005. "Empirical Bayes estimation by wavelet series," Statistics & Risk Modeling, De Gruyter, vol. 23(3), pages 181-198, March.
    2. Pensky Marianna, 2002. "A New Approach To Empirical Bayes Estimation With Errors In Variables," Statistics & Risk Modeling, De Gruyter, vol. 20(1-4), pages 225-240, April.

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