Abstract
Let \(X_1,\ldots ,X_n\) be a random sample from a distribution function \(F\) that denote lifetimes of \(n\) components of a coherent system. Suppose the system fails at \(X_{k:n}\), the \(k\)th order statistic of \(X\)’s, since we are not aware of the exact time at which the system has been failed, the residual lifetimes of the remaining \(n-k\) components, denoted by \(X^{(k)}_1,\ldots ,X^{(k)}_{n-k}\), are no longer independent but exchangeable. In this paper, multivariate stochastic comparisons of two vectors of lifetimes of the remaining components in the two sample problems are studied. Some sufficient conditions under which multivariate mixture models are compared stochastically with respect to the multivariate likelihood ratio ordering, the multivariate hazard rate ordering and the multivariate reversed hazard rate ordering are provided. These comparisons are done for different choices of the mixed distributions as well as mixing distributions. The new results obtained are applied to compare multivariate mixtures of location models.
Similar content being viewed by others
References
Bairamov I, Arnold BC (2008) On the residual lifelengths of the remaining components in a (n-k+1)- out-of-n system. Stat Probab Lett 78:945–952
Balakrishnan N, Barmalzan G, Haidari A (2014) Stochastic orderings and ageing properties of residual life lengths of live components in \((n-k+1)\)-out-of-\(n\) systems. J Appl Prob 51:58–68
Belzunce F (2013) Multivariate comparisons of ordered data. In: Li H, Li X (eds) Stochastic orders in reliability and risk. Springer, New York, pp 83–102
Deshpande JV, Singh H, Bagai I, Jain K (1990) Some partial orders describing positive ageing. Stoch Models 6:471–482
Dewan I, Khaledi B-E (2014) On stochastic comparisons of residual life time at random time. Stat Probab Lett 88:73–79
Hu T, Khaledi B-E, Shaked M (2003) Multivariate hazard rate orders. J Multivariate Anal 84:173–189
Johnson NL, Kotz S (1975) A vector valued multivariate hazard rate. J Multivariate Anal 5:53–66
Karlin S, Rinott Y (1980) Classes of orderings of measures and related correlation inequalities. I: multivariate totally positive distributions. J Multivariate Anal 10:467–498
Khaledi B-E, Shaked M (2010) Stochastic comparisons of multivariate mixtures. J Multivariate Anal 101:2486–2498
Khaledi B-E, Kochar S (2001) Dependence properties of multivariate mixture distributions and their applications. Ann Inst Statist Math 53:620–630
Kochar S (1999) On stochastic orderings between distributions and their sample spacings. Stat Probab Lett 42:345–352
Marshall AW (1975) Some comments on the hazard gradient. Stoch Proce Appl 3:293–300
Misra AK, Misra N (2012) Stochastic properties of conditionally independent mixture models. J Stat Plan Infer 142:1599–1607
Nanda AK, Bhattacharjee S, Balakrishnan N (2010) Mean residual life function, associated orderings and properties. IEEE Trans Reliab 59:55–65
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York
Acknowledgments
The authors are grateful to the Editor and the anonymous referees for their valuable comments and careful reading which have improved the presentation and the contents of the paper. The research of Baha-Eldin Khaledi partially supported by Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amini-Seresht, E., Khaledi, BE. Multivariate stochastic comparisons of mixture models. Metrika 78, 1015–1034 (2015). https://doi.org/10.1007/s00184-015-0538-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-015-0538-8
Keywords
- Hazard gradient
- Log-concave
- Log-convex and mean residual life functions
- Mean residual life order
- Multivariate stochastic orders
- \(MTP_2\) and \(TP_2\) functions