Bayesian Analysis of the Rayleigh Paired Comparison Model under Loss Functions using Informative Prior

Document Type : Research Note

Authors

1 Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan

2 Department of Statistics, Quaid-i-Azam University, Islamabad,Pakistan

Abstract

A number of paired comparison (PC) models exists in the literature. In this paper, the posterior distribution of the parameters of the Rayleigh PC model is derived using informative prior: Conjugate and Dirichlet. The values of the hyperparameters are elicited using prior predictive distribution. The preferences for the data of cigarette brands: Goldleaf (GL), Marlboro (ML), Dunhill (DH) and Benson & Hedges (BH) are collected from university students. The posterior estimates of the parameters are obtained under the loss functions: Quadratic Loss Function (QLS), Weighted Loss Function (WLS)and Squared Error Loss Function (SELF) with their risks. The preference and predictive probabilities are calculated. The posterior probabilities, for the hypothesis of comparing two parameters are evaluated. The graphs of marginal posterior distributions are given. Appropriateness of the model is tested by Chi-Square.

Keywords

Main Subjects

References

References

1. Aslam, M. An application of prior predictive distribution to elicit the prior density", Journal of Statistical Theory and Applications, 2(1), pp. 70-83 (2003).

2. Cattelan, M. Models for paired comparison data: A review with emphasis on dependent data", Statistical Science, 27(3), p. 412433 (2012).
3. Chaloner, K.M. and Duncan, G.T. Assessment of a beta prior distribution: PM elicitation", The Statistician, 32(1/2), pp. 174-180 (1983).
4. Gavasakar, U. A comparison of two elicitation methods for a prior distribution for a binomial parameter",
Management Science, 34(6), pp. 784-790 (1998).
5. Kadane, J.B. and Wolfson, L.J. Experiences in elicitation",
The Statistician, 47(1), pp. 3-19 (1998).
6. Liu, K.H. and Shih, Y.S. Score-scale decision tree
for paired comparison data", Statistica Sinica, 26, pp.
429-444 (2016).
7. Masarotto, G. and Varin, C. The ranking Lasso and
its application to sport tournaments", The Annals of
Applied Statistics, 6(4), pp. 1949-1970 (2012).
8. Pankratova, N. and Nedashkovskaya, N. The method
of estimating the consistency of paired comparisons",
International Journal Information Technologies &
Knowledge, 7(4), pp. 347-361 (2013).
9. Tutz, G. and Schauberger, G. Extended ordered
paired comparison models with application to football
data from German Bundesliga", AStA Advances in
Statistical Analysis, 99(2), pp. 209-227 (2015).
10. Veghes, C. Using the paired comparison method in
improving soccer rankings: The case of the Romanian
rst division", Annales Universitatis Apulensis Series
Oeconomica, 16(2), pp. 379-388 (2014).
11. Yan, T., Yang, Y., and Xu, J. Sparse paired comparisons
in the Bradley-Terry model", Statistica Sinica,
22, pp. 1305-1318 (2012).
12. Stern, H. A continuum of paired comparisons models",
Biometrika, 77(2), pp. 265-273 (1990).
13. Rai a, H. and Schlaifer, R., Applied Statistical Decision
Theory, Division of Research, Harvard Business
School, Boston (1961).
990 M. Aslam and T. Kifayat/Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 983{990
14. Je reys, H., Theory of Probability, Oxford University
Press, Oxford UK (1961).
15. Abbas, N. and Aslam, M. Prioritizing the items
through paired comparison models, a Bayesian approach",
Pakistan Journal of Statistics, 25(1), pp. 59-
69 (2009).

Scientia Iranica
Volume 25, Issue 2
Transactions on Industrial Engineering (E)
March and April 2018
Pages 983-990

Files

History

  • Receive Date: 29 March 2015
  • Revise Date: 25 November 2016
  • Accept Date: 06 March 2017

Share

How to cite

Statistics