Abstract
Most studies on reliability analysis have been conducted in homogeneous populations. However, homogeneous populations can rarely be found in the real world. Populations with specific components, such as lifetime, are usually heterogeneous. When populations are heterogeneous, it raises the question of whether these different modeling analysis strategies might be appropriate and which one of them should be preferred. In this paper, we provide mixture models, which have usually been effective tools for modeling heterogeneity in populations. Specifically, we carry out a stochastic comparison of two arithmetic (finite) mixture models using the majorization concept in the sense of the usual stochastic order, the hazard rate order, the reversed hazard rate order and the dispersive order both for a general case and for some semiparametric families of distributions. Moreover, we obtain sufficient conditions to compare two geometric mixture models. To illustrate the theoretical findings, some relevant examples and counterexamples are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albabtain AA, Shrahili M, Al-Shehri MA, Kayid M (2020) Stochastic comparisons of weighted distributions and their mixtures. Entropy 22(8):843
Amini-Seresht E, Zhang Y (2017) Stochastic comparisons on two finite mixture models. Oper Res Lett 45(5):475–480
Asadi M, Ebrahimi N, Soofi ES (2019) The alpha-mixture of survival functions. J Appl Probab 56(4):1151–1167
Badia FG, Lee H (2020) On stochastic comparisons and ageing properties of multivariate proportional hazard rate mixtures. Metrika 83(3):355–375
Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing: probability models. Florida State University Tallahassee
Bartoszewicz J (1987) A note on dispersive ordering defined by hazard functions. Stat Probab Lett 6:13–16
Block HW, Savits TH, Wondmagegnehu ET (2003) Mixtures of distributions with increasing linear failure rates. J Appl Probab 40:485–504
Cha JH, Finkelstein M (2013) The failure rate dynamics in heterogeneous populations. Reliab Eng Syst Saf 112:120–128
Davis DJF (1952) An analysis of some failure data. J Am Stat Assoc 47(258):113–150
Everitt BS, Hand DJ (1981) Finite mixture distributions. Chapman and Hall, New York
Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, Berlin
Hazra NK, Finkelstein M (2018) On stochastic comparisons of finite mixtures for some semiparametric families of distributions. TEST 27:988–1006
Li Y, Gu XM, Zhao J (2018) The weighted arithmetic mean-geometric mean inequality is equivalent to the Hölder inequality. Symmetry 10(9):380
Marshall AW, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its applications. Springer, New York
Mirhossaini SM, Dolati A (2008) On a new generalization of the exponential distribution. J Math Ext 3(1):27–42
Misra N, Naqvi S (2018) Stochastic comparison of residual lifetime mixture models. Oper Res Lett 46(1):122–127
Nadeb H, Torabi H (2020) New results on stochastic comparisons of finite mixtures for some families of distributions. Commun Stat Theory Methods 2:1–16
Navarro J (2008) Likelihood ratio ordering of order statistics, mixtures and systems. J Stat Plan Inference 138(5):1242–1257
Navarro J (2016) Stochastic comparisons of generalized mixtures and coherent systems. TEST 25(1):150–169
Navarro J, del Aguila Y (2017) Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components. Metrika 80(6–8):627–648
Navarro J, Del Águila Y, Suárez-Llorens SMAA (2016) Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Methodol Comput Appl Probab 18(2):529–545
Shaw WT, Buckley IR (2009). The alchemy of probability distributions: beyond GramCharlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. arXiv PreprintarXiv:0901.0434
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, Berlin
Shojaee O, Asadi M, Finkelstein M (2021) On some properties of \(\alpha \)-mixtures. Metrika 84(8):1–28
Shojaee O, Asadi M, Finkelstein M (2021) Stochastic properties of generalized finite \(\alpha \)-mixtures. Probab Eng Inform Sci 5:1–25
Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. Wiely, New York
Acknowledgements
The authors thank an associate editor and anonymous reviewers for their constructive comments which led to improve the contributions and exposition of this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shojaee, O., Babanezhad, M. On some stochastic comparisons of arithmetic and geometric mixture models. Metrika 86, 499–515 (2023). https://doi.org/10.1007/s00184-022-00880-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-022-00880-3