Ordering results of extreme order statistics from multiple-outlier scale models with dependence

Document Type : Original Article

Authors

1 Theoretical Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, India

2 Department of Mathematics, National Institute of Technology Rourkela, India

Abstract

Many authors have studied ordering results between extreme order statistics from multiple-outlier models when the observations are independent. However, the independence assumption is not very attractive in many situations due to the complexity of the problems. This paper focuses on stochastic comparisons of extreme order statistics stemming from multiple-outlier scale models with dependence. Archimedean copula is used to model dependence structure among nonnegative random variables. Sufficient conditions are obtained to compare the largest order statistics in the sense of the usual stochastic, reversed hazard rate, likelihood ratio, dispersive, star, and Lorenz orders. The smallest order statistics are also compared with respect to the usual stochastic, hazard rate, star, and Lorenz orders. Here, the sufficient conditions are based on the weak-super majorization, weak-sub majorization, and p-larger orders between the model parameters.  To illustrate the theoretical establishments, some examples are provided. Furthermore, some counterexamples are provided to establish that ignorance of sufficient conditions may not lead to the established ordering results between the order statistics.

Keywords


  1.  Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Advances in Applied Probability, 18, 1019–1022.
  2. Amini-Seresht, E., Qiao, J., Zhang, Y. and Zhao, P. (2016). On the skewness of order statistics in multiple-outlier PHR models. Metrika, 79, 817–836.
  3.  Balakrishnan, N. and Torrado, N. (2016). Comparisons between largest order statistics from multiple-outlier models. Statistics, 50, 176–189.
  4. Balakrishnan, N. and Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences, 27, 403–443.
  5. Fang, L., Barmalzan, G. and Ling, J. (2016). Dispersive order of lifetimes of series systems in multiple-outlier Weibull models. Journal of Systems Science & Complexity, 29, 1693–1702.
  6. Kochar, S. C. and Torrado, N. (2015). On stochastic comparisons of largest order statistics in the scale model. Communications in Statistics-Theory and Methods, 44, 4132–4143.
  7. Kochar, S. and Xu, M. (2011). On the skewness of order statistics in multiple-outlier models. Journal of Applied Probability, 48, 271–284.
  8. Kundu, A., Chowdhury, S., Nanda, A. K. and Hazra, N. K. (2016). Some results on majorization and their applications. Journal of Computational and Applied Mathematics, 301, 161–177.
  9. Li, C., Fang, R. and Li, X. (2016). Stochastic comparisons of order statistics from scaled and interdependent random variables. Metrika, 79, 553–578.
  10. Li, X. and Fang, R. (2015). Ordering properties of order statistics from random variables of Archimedean copulas with applications. Journal of Multivariate Analysis, 133, 304–320.
  11. Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.
  12. Marshall, A.W., Olkin, I. and Arnold, B.C. (2011). Inequalities: Theory of Majorization and its Applications. Second edn, Springer, New York.
  13. McNeil, A. J. and Nešlehová, J. (2009). Multivariate archimedean copulas, d-monotone functions and l1-norm symmetric distributions. The Annals of Statistics, 37, 3059–3097.
  14. Navarro, J., Torrado, N. and Águila, Y.d. (2018). Comparisons between largest order statistics from multiple-outlier models with dependence. Methodology and Computing in Applied Probability, 20, 411–433.
  15. Nelsen, R. B. (2006) An Introduction to Copulas. Second edn. Springer, New York.
  16. Saunders, I. W. and Moran, P. A. P. (1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability, 15, 426–432.
  17. Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. Springer, New York.
  18. Torrado, N. (2017). Stochastic comparisons between extreme order statistics from scale models. Statistics, 51, 1359–1376.
  19. Wang, J. and Cheng, B. (2017). Answer to an open problem on mean residual life ordering of parallel systems under multiple-outlier exponential models. Statistics & Probability Letters, 130, 80–84.
  20.  Zhang, Y., Cai, X., Zhao, P. and Wang, H. (2019). Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components. Statistics, 53, 126–147.
  21. Zhao, P. and Balakrishnan, N. (2012). Stochastic comparisons of largest order statistics from multiple-outlier exponential models. Probability in the Engineering and Informational Sciences, 26, 159–182.
  22. Zhao, P. and Balakrishnan, N. (2015). Comparisons of largest order statistics from multipleoutlier gamma models. Methodology and Computing in Applied Probability, 17, 617–645.
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