## Electronic Journal of Statistics

### A note on Dempster-Shafer recombination of confidence distributions

#### Abstract

It is often the case that there are several studies measuring the same parameter. Naturally, it is of interest to provide a systematic way to combine the information from these studies. Examples of such situations include clinical trials, key comparison trials and other problems of practical importance. Singh et al. (2005) provide a compelling framework for combining information from multiple sources using the framework of confidence distributions. In this paper we investigate the feasibility of using the Dempster-Shafer recombination rule on this problem. We derive a practical combination rule and show that under assumption of asymptotic normality, the Dempster-Shafer combined confidence distribution is asymptotically equivalent to one of the method proposed in Singh et al. (2005). Numerical studies and comparisons for the common mean problem and the odds ratio in $2\times 2$ tables are included.

#### Article information

Source
Electron. J. Statist., Volume 6 (2012), 1943-1966.

Dates
First available in Project Euclid: 12 October 2012

https://projecteuclid.org/euclid.ejs/1350046859

Digital Object Identifier
doi:10.1214/12-EJS734

Mathematical Reviews number (MathSciNet)
MR2988470

Zentralblatt MATH identifier
1295.62005

Subjects
Primary: 62A01: Foundations and philosophical topics
Secondary: 62F99: None of the above, but in this section

#### Citation

Hannig, Jan; Xie, Min-ge. A note on Dempster-Shafer recombination of confidence distributions. Electron. J. Statist. 6 (2012), 1943--1966. doi:10.1214/12-EJS734. https://projecteuclid.org/euclid.ejs/1350046859

#### References

• Bender, R., Berg, G. and Zeeb, H. (2005) Tutorial: Using confidence curves in medical research., Biometrical Journal, 47, 237–247.
• Birnbaum, A. (1961) Confidence curves: An omnibus technique for estimation and testing statistical hypotheses., J. Amer. Statist. Assoc., 56, 246–249.
• Casella, G. and Berger, R. L. (2002), Statistical Inference. Pacific Grove, CA: Wadsworth and Brooks/Cole Advanced Books and Software, 2nd edn.
• Cox, D. (1958) Some problems with statistical inference., The Annals of Mathematical Statistics, 29, 357–372.
• Dempster, A. P. (2008) The Dempster-Shafer Calculus for Statisticians., International Journal of Approximate Reasoning, 48, 365–377.
• Durrett, R. (2005), Probability: theory and examples. Duxburry Advanced Series. Brooks/Cole, third edn.
• Efron, B. (1993) Bayes and likelihood calculations from confidence intervals., Biometrika, 80, 3–26.
• Efron, B. (1998) R. A. Fisher in the 21st century., Statist. Sci., 13, 95–122. With comments and a rejoinder by the author.
• Fisher, R. (1973), Statistical Methods and Scientific Inference (3rd edition). New York: Hafner Press.
• Hannig, J. (2009) On Generalized Fiducial Inference., Statistica Sinica, 19, 491–544.
• Hannig, J. (2012) Generalized Fiducial Inference via Discretizion., Statistica Sinica.
• Hannig, J., Iyer, H. K. and Patterson, P. (2006) Fiducial generalized confidence intervals., Journal of American Statistical Association, 101, 254–269.
• Liu, D., Liu, R. and Xie, M. (2011) Exact meta-analysis approach for the common odds ratio of 2 by 2 tables with rare events., Technical Report, Department of Statistics, Rutgers University.
• Normand, S.-L. (1999) Meta-analysis: formulating, evaluating, combining, and reporting., Statistics in Medicine, 18, 321–359.
• Schweder, T. and Hjort, N. L. (2002) Confidence and likelihood., Scand. J. Statist., 29, 309–332. Large structured models in applied sciences; challenges for statistics (Grimstad, 2000).
• Shafer, G. (1976), A mathematical theory of evidence. Princeton, New Jersey: Princeton University Press.
• Singh, K., Xie, M. and Strawderman, W. E. (2001) Confidence distributions – concept, theory and applications., Tech. rep., Department of Statistics, Rutgers University. Updated 2004.
• Singh, K., Xie, M. and Strawderman, W. E. (2005) Combining information from independent sources through confidence distributions., The Annals of Statistics, 33, 159–183.
• Singh, K., Xie, M. and Strawderman, W. E. (2007) Confidence distribution (cd)-distribution estimator of a parameter., IMS Lecture Notes-Monograph Series, 54, 132–150.
• Strawderman, W. and Rukhin, A. (2010) Simultaneous estimation and reduction of nonconformity in inter-laboratory studies., J. R. Statist. Soc. B, 72, 219–234.
• Tian, L., Cai, T., Pfeffer, M. A., Piankov, N., Cremieux, P. Y. and Wei, L. J. (2009) Exact and efficient inference procedure for meta-analysis and its application to the analysis of independent $2\times 2$ tables with all available data but without artificial continuity correction., Biostatistics, 10, 275–281.
• Wang, J. C.-M., Hannig, J. and Iyer, H. K. a. (2012) Fiducial prediction intervals., Journal of Statistical Planning and Inference, 142, 1980–1990.
• Webb, K. S., Carter, D. and Wolff Briche, C. S. J. (2003) Ccqm-k21: Key comparison of the determination of pp$'$-ddt in fish oil, final report., Metrologia, 40, tech. suppl. 08004.
• Xie, M. and Singh, K. (2012) Confidence distribution, the frequentist distribution estimator of a parameter – a review., International Statistical Reviews. To appear (Invited review article with discussions).
• Xie, M., Singh, K. and Strawderman, W. E. (2011) Confidence distributions and a unified framework for meta-analysis., Journal of the American Statistical Association, 106, 320–333.
• Zhang, J. and Liu, C. (2011) Dempster-Shafer inference with weak beliefs., Statistica Sinica, 21, 475–494.