The Impact of Prior Based Loss Function for Elliptical Regression Models

Year 2024, Volume: 9 Issue: 2, 551 - 571, 29.12.2024

Mohammad Arashı , Fatma Sevinç Kurnaz , Naushad Mamodekhan

https://doi.org/10.33484/sinopfbd.1485489

Abstract

In the paper we consider a multiple regression model with elliptically contoured errors. In the Bayesian view, a prior information is taken for the weight under a prior based balanced-type loss function in order to avoid making redundant assumptions. This is the essence of the Bayesian inference with vague prior information in regression analysis. It directly impacts on the performance of the quasi empirical Bayesian shrinkage estimators through the inclusion of a reciprocal weight related to the dimension of parameter space. The shrinkage factor of the estimator is also robust to outliers and the unknown density generator of elliptical models. Finally, this result is supported by an application.

Keywords

Balanced loss function, Elliptically contoured distribution, Prior based loss function, Quasi- empirical Bayes estimator, Shrinkage estimator

Ethical Statement

The work does not require ethics committee approval and any private permission.

Supporting Institution

The authors have no received any financial support for the research, authorship, or publication of this study.

Thanks

We would like to thank two anonymous reviewers for their constructive comments that led us to add many details and improve the presentation.

Eliptik Regresyon Modelleri için Kayıp Fonksiyona Dayalı Önselliğin Etkisi

Abstract

Bu çalışmada eliptik konturlu hatalara sahip olan çoklu regresyon modeli ele alınmıştır. Bayesyen bakıştan, gereksiz varsayımlarda bulunmaktan kaçınmak için, ağırlık için dengeli tipteki kayıp fonksiyonuna dayalı bir önsellik altında önsellik bilgisi gözönüne alınmıştır. Bu, regresyon analizindeki muğlak öncelik bilgisiyle Bayesyen çıkarımın özüdür. Parametre uzayının boyutuyla ilgili karşılıklı ağırlığın dahil edilmesi yoluyla yarı ampirik Bayesyen büzücü tahmincilerin performansı bundan etkilenir. Tahmin edicinin büzülme faktörünün, verilerdeki aykırı değerlere ve eliptik modellerin bilinmeyen yoğunluk yaratıcı fonksiyonuna karşı dayanıklı olduğu gösterilmiştir. Son olarak, bir uygulama ile desteklenmiştir.

Keywords

Dengeli kayıp fonksiyonu, Eliptik konturlu dag ̆ılım, Kayıp fonksiyona dayalı önce- lik, Yarı-ampirik Bayes tahmin edici, Büzücü tahmin edici

References

• Fang, K. T., & Zhang, Y. T. (1990). Generalized Multivariate Analysis. Springer, New York.
• Jozani, M. J., Marchand, E., & Parsian, A. (2006). On estimation with weighted balanced-type loss function. Statistics and Probability Letters, 76, 773–780. https://doi.org/10.1016/j.spl.2005.10.026
• Zellner, A.(1994). Bayesian and non-Bayesian estimation using balanced loss functions. In: J.O. Berger and S.S. Gupta, (Eds.), Statistical Decision Theory and Methods V, Springer, New York, pp. 337–390.
• Gomez-Deniz, E. (2008). A generalization of the credibility theory obtained by using the weighted balanced loss function. Insurance: Mathematics and Economics, 42, 850-854. https://doi.org/10.1016/j.insmatheco.2007.09.002
• Dey, D., Ghosh, M., & Strawderman, W. E. (1999). On estimation with balanced loss function. Statistics and Probability Letters, 45, 97–101. https://doi.org/10.1016/S0167-7152(99)00047-4
• Evans, M., & Jang, G. H. (2011). Inference from prior-based loss function. Technical report No. 1104, University of Toronto.
• Saleh, A. K. Md. E. (2006). Theory of Preliminary Test and Stein-type Estimation with Applications. John Wiley, New York.
• Arashi, M., Saleh, A. K., Ehsanes, A. K. Md., & Tabatabaey, S. M. M. (2013). Regression model with elliptically contoured errors. Statistics, 47(6), 1266–1284. https://doi.org/10.1080/02331888.2012.694442
• Cellier, D., Fourdrinier, D., and Strawderman, W. E. (1995). Shrinkage positive rule estimators for spherically symmetrical distributions. Journal of multivariate analysis, 532, 194–209. https://doi.org/10.1006/jmva.1995.1032
• Khan, S. & Saleh, A. K. Md. E. (1997). Shrinkage Pre-Test Estimator of the Intercept Parameter for a Regression Model with Multivariate Student-t Errors. Biometrical Journal, 392, 131–147. https://doi.org/10.1002/bimj.4710390202
• Khan, S. & Saleh, A. K. Md. E. (1998). Comparison of estimators of means based on p-samples from multivariate student-t population. Communications in Statistics-Theory and Methods, 271, 193–210. https://doi.org/10.1080/03610929808832660
• Khan, S. & Saleh, A. K. Md. E. (2001). On the comparison of the pre-test and shrinkage estimators for the univariate normal mean. Statistical Papers, 42(4), 451–473.
• Khan, S., Hoque, Z., & Saleh, A. K. Md. E. (2002). Estimation of the slope parameter for linear regression model with uncertain prior information. Journal of Statistical Research, 36, 55–73.
• Khan, S. (2005). Estimation of parameters fo the simple multivariate linear models with Student-t errors. Journal of Statistical Research,, 39(2), 71–86.
• Khan, S. (2008). Shrinkage Estimators of Intercept Parameters of Two Simple Regression Models with Suspected Equal Slopes. Communications in Statistics—Theory and Methods, 37, 247–260.https://doi.org/10.1080/03610920701648961
• Saleh, A. K. Md. E & Kibria, B. M. G. (2011). On Some Ridge Regression Estimators: A Nonparametric Approach. Journal of Nonparametric Statistics, 233, 819–851. https://doi.org/10.1080/10485252.2011.567335
• Saleh, A. K. Md. E & Sen, P. K. (1978). Nonparametric Estimation of Location Parameters after a Preliminary Test on Regression. The Annals of Statistics, 6, 154–168.
• Stein, C. (1956). The admissibility of Hotelling’s T2-test. The Annals of Mathematical Statistics, 27, 616–623.
• Tabatabaey, S. M. M., Saleh, A. K. Md. E.,& Kibria, B. M. G. (2004). Simultaneous estimation of regression parameters with spherically symmetric errors under possible stochastic constrains. International Journal of Statistical Sciences, 3, 1–20.
• Jeffreys, H. (1961). Theory of Probability, Oxford: Clarendon, 1961.
• Anderson, T. W., Fang, K. T., & Hsu, H. (1986). Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions. The Canadian Journal of Statistics, 14, 55–59. https://doi.org/10.2307/3315036
• Arashi, M. (2012). Preliminary test and Stein estimators in simultaneous linear equations. Linear Algebra and its Applications, 436(5), 1195–1211. https://doi.org/10.1016/j.laa.2011.07.036
• Srivastava, M., & Bilodeau, M. (1989). Stein estimation under elliptical distribution. Journal of Multivariate Analysis, 28, 247–259. https://doi.org/10.1016/0047-259X(89)90108-5
• Khan, S. (2000). Improved estimation of the mean vector for student-t model. Communications in Statistics-Theory and Methods, 293, 507–527. https://doi.org/10.1080/03610920008832499
• Arashi, M., Saleh, A. K. Md. E., & Tabatabaey, S. M. M. (2010). Estimation of parameters of parallelism model with elliptically distributed errors. Metrika, 71, 79-100. https://doi.org/10.1007/s00184-008-0203-6
• Saleh, A. K. Md. E., Arashi, M., & Tabatabaey, S. M. M. (2014). Statistical Inference for Models with Multivariate t-Distributed Errors, John Wiley, New Jersey.