In recent years, we have studied information properties of various types of mixtures of probability distributions and introduced a new type, which includes previously known mixtures as special cases. These studies are disseminated in different fields: reliability engineering, econometrics, operations research, probability, the information theory, and data mining. This paper presents a holistic view of these studies and provides further insights and examples. We note that the insightful probabilistic formulation of the mixing parameters stipulated by Behboodian (1972) is required for a representation of the well-known information measure of the arithmetic mixture. Applications of this information measure presented in this paper include lifetime modeling, system reliability, measuring uncertainty and disagreement of forecasters, probability modeling with partial information, and information loss of kernel estimation. Probabilistic formulations of the mixing weights for various types of mixtures provide the Bayes-Fisher information and the Bayes risk of the mean residual function.

Type of Study: Special Issue, Original Paper |
Subject:
62Exx: Distribution theory

Received: 2021/02/10 | Accepted: 2021/02/23 | Published: 2021/06/20

Received: 2021/02/10 | Accepted: 2021/02/23 | Published: 2021/06/20

1. Ardakani, O. M., Ebrahimi, N., and Soofi, E. S. (2018), Ranking forecasts by stochastic error distance, information and reliability measures. International Statistical Review, 86(3), 442-468. [DOI:10.1111/insr.12250]

2. Ardakani, O. M., Asadi, M., Ebrahimi, N., and Soofi, E. S. (2020), MR plot: A big data tool for distinguishing distributions. Statistical Analysis and Data Mining the ASA Data Science Journal, 13, 405-418. [DOI:10.1002/sam.11464]

3. Asadi, M. and Zohrevand, Y. (2007), On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference, 137, 1931-1941. [DOI:10.1016/j.jspi.2006.06.035]

4. Asadi, M., Ebrahimi, N., and Soofi, E. S. (2017), Connections of Gini, Fisher, and Shannon, by Bayes risk under proportional hazards. Journal of Applied Probability, 54, 1027-1050. [DOI:10.1017/jpr.2017.51]

5. Asadi, M., Ebrahimi, N., and Soofi, E. S. (2018), Optimal hazard models based on partial information. European Journal of Operational Research, 270(2), 1-11. [DOI:10.1016/j.ejor.2018.04.006]

6. Asadi, M., Ebrahimi, N., and Soofi, E. S. (2019), The alpha-mixture of survival functions. Journal of Applied Probability, 56(4), 1151-1167. [DOI:10.1017/jpr.2019.72]

7. Asadi, M., Ebrahimi, N., kharazmi, O., and Soofi, E. S. (2019), Mixture models, Bayes Fisher information, and divergence measures. IEEE Transactions on Information Theory, 65, 2316-2321. [DOI:10.1109/TIT.2018.2877608]

8. Asadi, M., Ebrahimi, N., Soofi, E. S., and Zohrevand, Y. (2016), Jensen-Shannon information of the coherent system lifetime. Reliability Engineering and System Safety, 156(C), 244-255. [DOI:10.1016/j.ress.2016.07.015]

9. Bajgiran, A. H., Mardikoraem, M., and Soofi, E. S. (2021), Maximum entropy distributions with quantile information. European Journal of Operational Research, 290(1), 196-209. [DOI:10.1016/j.ejor.2020.07.052]

10. Barlow, R. E., Marshall, A. W., and Proschan, F. (1963), Properties of probability distributions with monotone hazard rate. Annals of Mathematical Statistics, 34, 375-389. [DOI:10.1214/aoms/1177704147]

11. Behboodian, J. (1970), On a mixture of normal distributions. Biometrika, 57(1), 215-217. [DOI:10.1093/biomet/57.1.215]

12. Behboodian, J. (1972), On the distribution of a symmetric statistics from a mixed population. Technometrics, 14, 919-923. [DOI:10.1080/00401706.1972.10488987]

13. Beheshti, N., Racine, J. S., and Soofi, E. S. (2019), Information measures of kernel estimation. Econometric Reviews, 38(1), 47-68. [DOI:10.1080/07474938.2016.1222236]

14. Bercher, J. F. (2011), Escort entropies and divergences and related canonical distribution. Physics Letters A, 375, 2969-2973. [DOI:10.1016/j.physleta.2011.06.057]

15. Bercher, J. F. (2012), A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians. Physica a: Statistical Mechanics and Its Applications, 391(19), 4460-4469. [DOI:10.1016/j.physa.2012.04.024]

16. Bissiri, P. G., Holmes, C. C., and Walker, S. G. (2016), A general framework for updating belief distributions. Journal of the Royal Statistical Society Series B, 78(5), 1103-1130. [DOI:10.1111/rssb.12158]

17. Block, H. W., and Savits, T. H. (1997). Burn-in. Statistical Science, 12, 1-19. [DOI:10.1214/ss/1029963258]

18. Chernoff, H. (1952), A measure of asymptotic efficiency of tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics, 23 ,493-507. [DOI:10.1214/aoms/1177729330]

19. Cover, T. M. and Thomas, J. A. (2006), Elements of Information Theory, 2nd ed.. New York: Wiley.

20. Harremoes, P. (2001), Binomial and Poisson distributions as maximum entropy distributions. IEEE Transactions on Information Theory, 47, 2039-2041. [DOI:10.1109/18.930936]

21. Holmes, C. C., and Walker, S. G. (2017). Assigning a value to a power likelihood in a general Bayesian model. Biometrika, 104, 497-503.

22. Ibrahim, J. G., and Chen, M. H. (2000), Power prior distributions for regression models. Statistical Science, 15, 46-60.

23. Ibrahim, J. G., Chen, M. H., and Sinha, D. (2003), On optimality of the power prior. Journal of the American Statistical Association, 98, 204-213. [DOI:10.1198/016214503388619229]

24. Kochar, S., Mukerjee, H., and Samaniego, F. J. (1999), The "signature" of a coherent system and its application to comparisons among systems. Naval Research Logistics, 46(5), 507-523.
https://doi.org/10.1002/(SICI)1520-6750(199908)46:5<507::AID-NAV4>3.0.CO;2-D [DOI:10.1002/(SICI)1520-6750(199908)46:53.0.CO;2-D]

25. Kullback, S. (1959), Information theory and statistics. New York: Wiley (reprinted in 1968 by Dover).

26. Li, Q., and Racine (2007). Nonparametric Econometrics: Theory and Practice. New Jersey: Princeton University Press.

27. Lin, J. (1991), Divergence measures based on the Shannon entropy. Transactions on Information Theory, 37, 145-151. [DOI:10.1109/18.61115]

28. Lindley, D. V. (1956), On a measure of the information provided by an experiment. Annals of Mathematical Statistics, 27, 986-1005. [DOI:10.1214/aoms/1177728069]

29. Lynn, N. J., and Singpurwalla, N. D. (1997), Comment: "Burn-in'" makes us feel good. Statistical Science, 12, 13-19.

30. McCulloch, R. E. (1989), Local model influence. Journal of the American Statistical Association, 84, 473-478. [DOI:10.1080/01621459.1989.10478793]

31. Mcvinish, M., Rousseau, J., and Mengersen, K. (2009), Bayesian goodness of fit testing with mixtures of triangular distributions. Scand. J Statist., 36, 337-354.

32. Navarro, J., and Rychlik, T. (2007), Reliability and expectation bounds for coherent systems with exchangeable components. Journal of Multivariate Analysis, 98(1), 102-113. [DOI:10.1016/j.jmva.2005.09.003]

33. Rao, M., Chen, Y., Vemuri, B. C., and Wang, F. (2004), Cumulative residual entropy: A new measure of information. IEEE Transactions on Information Theory, 50, 1220-1228. [DOI:10.1109/TIT.2004.828057]

34. Samaniego, F. J. (1985), On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability, R-34(1), 69-72. [DOI:10.1109/TR.1985.5221935]

35. Samaniego, F. J. (2007), System signatures and their applications in engineering reliability. Springer.

36. Shaked, M., and Shanthikumar, J. G. (2007), Stochastic orders. Springer.

37. Shaked, M., and Suarez-Llorens, A. (2003), On the comparison of reliability experiments based on the convolution order. Journal of the American Statistical Association, 98(463), 693-702. [DOI:10.1198/016214503000000602]

38. Shoja, M., and Soofi, E. S. (2017), Uncertainty, information, and disagreement of economic forecasters. Econometric Reviews, 36(6-9), 796-817. [DOI:10.1080/07474938.2017.1307577]

39. Soofi, E. S., Ebrahimi, N., and Habibullah, M. (1995), Information distinguishability with application to analysis of failure data. Journal of the American Statistical Association, 90, 657-668. [DOI:10.1080/01621459.1995.10476560]

40. Tsallis, C. (1998), Generalized entropy-based criterion for consistent testing. Physics Review E, 58, 1442-1445. [DOI:10.1103/PhysRevE.58.1442]

41. Vakili-Nezhaad, G. R., and Mansoori, G. A. (2004), An application of non-extensive statistical mechanics to nanosystems. Journal of Computational and Theortical Nanonscience, 1, 233-235. [DOI:10.1166/jctn.2004.021]

42. van Erven, T., and Harremo"es, P. (2014), R'enyi divergence and Kullback-Leibler divergence. IEEE Transactions on Information Theory, {bf 60, 3797-3820. [DOI:10.1109/TIT.2014.2320500]

43. Walker, S. G. (2016), Bayesian information in an experiment and the Fisher information distance. Statistics and Probability Letters, {bf 112, 5-9. [DOI:10.1016/j.spl.2016.01.014]

44. Wang, W., and Lahiri, K. (2021), Estimating macroeconomic uncertainty and discord using info-metrics. In Innovations in info-metrics A cross-disciplinary perspective on information and information processing, 1-55.

45. Wang, L., and Madiman, M. (2014), Beyond the entropy power inequality, via rearrangements. IEEE Transactions on Information Theory, 60(9), 5116-5137. [DOI:10.1109/TIT.2014.2338852]

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