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Optimal rules and robust Bayes estimation of a Gamma scale parameter

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  • Leila Golparver
  • Ali Karimnezhad
  • Ahmad Parsian

Abstract

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $$cX\,{+}\,d$$ , when $$X\sim \varGamma (\alpha ,\eta )$$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail. Copyright Springer-Verlag 2013

Suggested Citation

  • Leila Golparver & Ali Karimnezhad & Ahmad Parsian, 2013. "Optimal rules and robust Bayes estimation of a Gamma scale parameter," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(5), pages 595-622, July.
  • Handle: RePEc:spr:metrik:v:76:y:2013:i:5:p:595-622
    DOI: 10.1007/s00184-012-0407-7
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    References listed on IDEAS

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    1. Rukhin, Andrew L. & Ananda, Malwane M. A., 1992. "Risk behavior of variance estimators in multivariate normal distribution," Statistics & Probability Letters, Elsevier, vol. 13(2), pages 159-166, January.
    2. Jozani, Mohammad Jafari & Nematollahi, Nader & Shafie, Khalil, 2002. "An admissible minimax estimator of a bounded scale-parameter in a subclass of the exponential family under scale-invariant squared-error loss," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 437-444, December.
    3. Kiapour, A. & Nematollahi, N., 2011. "Robust Bayesian prediction and estimation under a squared log error loss function," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1717-1724, November.
    4. Boratynska, Agata, 1997. "Stability of Bayesian inference in exponential families," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 173-178, December.
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    Cited by:

    1. Ali Karimnezhad & Mahmoud Zarepour, 2020. "A general guide in Bayesian and robust Bayesian estimation using Dirichlet processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(3), pages 321-346, April.
    2. Jafar Ahmadi & Elham Mirfarah & Ahmad Parsian, 2016. "Robust Bayesian Pitman closeness," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(6), pages 671-691, August.
    3. Riyadh Rustam Al-Mosawi & Shahjahan Khan, 2018. "Estimating moments of a selected Pareto population under asymmetric scale invariant loss function," Statistical Papers, Springer, vol. 59(1), pages 183-198, March.

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