Abstract
For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form \({L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}\), as well as the weighted version \({q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}\), where ρ(θ, δ) is an arbitrary loss function, δ 0 is a chosen a priori “target” estimator of \({\theta, \omega \in[0,1)}\), and q(·) is a positive weight function. we develop Bayesian estimators under \({L_{\rho, \omega, \delta_0}}\) with ω > 0 by relating such estimators to Bayesian solutions under \({L_{\rho, \omega, \delta_0}}\) with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.
Similar content being viewed by others
References
Ahmadi J, Jafari Jozani M, Marchand É, Parsian A (2009a) Bayes estimation based on k-record data from a general class of distributions under balanced type loss functions. J Stat Plan Inference 139: 1180–1189
Ahmadi J, Jafari Jozani M, Marchand É, Parsian A (2009b) Prediction of k-records from a general class of distributions under balanced type loss functions. Metrika 70: 19–33
Bansal AK, Aggarwal P (2007) Bayes prediction for a heteroscedastic regression super-population model using balanced loss function. Commun Stat 36: 1565–1575
Berger JO (1985) Statistical decision theory and Bayesian analysis. Springer-Verlag, New York
Berger JO (1994) An overview of Robust Bayesian analysis. Test 3: 5–124
Betro B, Ruggeri F (1992) Conditional Γ-minimax actions under convex losses. Commun Stat 21: 1051–1066
Boratyńska A (1997) Stability of Bayesian inference in exponential families. Stat Probab Lett 36: 173–178
DasGupta A, Studden W (1991) Robust Bayesian experimental designs in normal linear models. Ann Stat 19: 1244–1256
Dey D, Ghosh M, Strawderman WE (1999) On estimation with balanced loss functions. Stat Probab Lett 45: 97–101
Ghosh M, Kim MJ, Kim D (2007) Constrained Bayes and empirical Bayes estimation with balanced loss function. Commun Stat 36: 1527–1535
Ghosh M, Kim MJ, Kim D (2008) Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function. J Stat Plan Inference 8: 2017–2028
Gómez-Déniz E (2008) A generalization of the credibility theory obtained by using the weighted balanced loss function. Insur Math Econ 42: 850–854
Heilmann W (1989) Decision theoretic foundations of credibility theory. Insur Math Econ 8: 77–95
Jafari Jozani M, Parsian A (2008) Posterior regret Γ-minimax estimation and prediction based on k-record data under entropy loss function. Commun Stat 37(14): 2202–2212
Jafari Jozani M, Marchand É, Parsian A (2006) On estimation with weighted balanced-type loss function. Stat Probab Lett 76: 773–780
Marchand É, Strawderman WE (2004) Estimation in restricted parameter spaces: a review. A Festschrift for Herman Rubin, IMS Lecture Notes-Monograph Series, vol 45. Institute of Mathematical Statistics, Hayward, California, pp 21–44
Meczarski M (1993) Stability and conditional Γ-minimaxity in Bayesian inference. Appl Math 22: 117–122
Meczarski M, Zielinski R (1991) Stability of Bayesian estimator of the Poisson mean under the inexactly specified Gamma prior. Stat Probab Lett 12: 329–333
Parsian A, Kirmani SNUA (2002) Estimation under LINEX loss function. Handbook of applied econometrics and statistical inference, Statistics textbooks monographs. Marcel Dekker Inc., pp 53–76
Rios Insua D, Ruggeri F (2000) Robust Bayesian analysis. Lecture notes in statistics. Springer-Verlag, New York
Rios Insua D, Ruggeri F, Vidakovic B (1995) Some results on posterior regret Γ-minimax estimation. Stat Decis 13: 315–351
Rodrigues J, Zellner A (1994) Weighted balanced loss function and estimation of the mean time to failure. Commun Stat 23: 3609–3616
Toutenburg H, Shalab (2005) Estimation of regression coefficients subject to exact linear restrictions when some observations are missing and quadratic error balanced loss function is used. Test 14: 385–396
Wolfe PJ, Godsill SJ (2003) A perceptually balanced loss function for short-time spectral amplitude estimation. Proc IEEE Conf 5: 425–428
Zellner A (1986) Bayesian estimation and prediction using asymmetric loss functions. J Am Stat Assoc 81: 446–545
Zellner A (1994) Bayesian and Non-Bayesian estimation using balanced loss functions. In: Berger JO, Gupta SS (eds) Statistical decision theory and methods, vol V. Springer-Verlag, New York, pp 337–390
Zen M, DasGupta A (1993) Estimating a binomial parameter: is robust Bayes real Bayes?. Stat Decis 11: 37–60
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jafari Jozani, M., Marchand, É. & Parsian, A. Bayesian and Robust Bayesian analysis under a general class of balanced loss functions. Stat Papers 53, 51–60 (2012). https://doi.org/10.1007/s00362-010-0307-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-010-0307-8
Keywords
- Balanced loss function
- Bayes estimator
- Robust Bayesian analysis
- Posterior risk
- Posterior regret gamma-minimax
- Conditional gamma-minimax
- Most stable estimation