Abstract
This paper is devoted to some ordering results for the largest and the smallest order statistics arising from dependent heterogeneous exponentiated location-scale random observations. We assume that the sets of observations are sharing a common or different Archimedean copula(s). Sufficient conditions for which the usual stochastic order and the reversed hazard rate order between the extreme order statistics hold are derived. Various numerical examples are provided for the illustration of the proposed results. Finally, some applications of the comparison results in engineering reliability and auction theory are presented.
Similar content being viewed by others
References
Balakrishnan N, Zhao P (2013) Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probab Eng Inf Sci 27(4):403–443
Balakrishnan N, Haidari A, Masoumifard K (2015) Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Trans Reliab 64(1):333–348
Balakrishnan N, Nanda P, Kayal S (2018) Ordering of series and parallel systems comprising heterogeneous generalized modified Weibull components. Appl Stoch Models Bus Ind 34(6):816–834
Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing: probability models. Holt, Rinehart and Winston, New York
Barmalzan G, Payandeh Najafabadi AT, Balakrishnan N (2017) Ordering properties of the smallest and largest claim amounts in a general scale model. Scand Actuar J 2017(2):105–124
Bartoszewicz J (1986) Dispersive ordering and the total time on test transformation. Stat Probab Lett 4(6):285–288
Bashkar E, Torabi H, Roozegar R (2017) Stochastic comparisons of extreme order statistics in the heterogeneous exponentiated scale model. J Stat Theory Appl 16(2):219–238
Bashkar E, Torabi H, Dolati A, Belzunce F (2018) \(f\)-Majorization with applications to stochastic comparison of extreme order statistics. J Stat Theory Appl 17(3):520–536
Boland PJ, Shaked M, Shanthikumar JG (1998) Stochastic ordering of order statistics. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 89–103
Boland PJ, Hu T, Shaked M, Shanthikumar JG (2005) Stochastic ordering of order statistics II. In: Modeling uncertainty, vol 46 of International series in operations research & management science. Springer, Boston, MA, pp 607–623
Chowdhury S, Kundu A (2017) Stochastic comparison of parallel systems with log-Lindley distributed components. Oper Res Lett 45(3):199–205
David H, Nagaraja H (2003) Order statistics, 3rd edn. Wiley, New York
Ding W, Zhang Y (2018) Relative ageing of series and parallel systems: effects of dependence and heterogeneity among components. Oper Res Lett 46(2):219–224
Dolati A, Towhidi M, Shekari M (2011) Stochastic and dependence comparisons between extreme order statistics in the case of proportional reversed hazard model. J Iran Stat Soc 10(1):29–43
Fang R, Li X, Yan R (2015) Impact of dependence among valuations on expected revenue in auctions. Am J Math Manag Sci 34(3):234–264
Fang R, Li C, Li X (2016) Stochastic comparisons on sample extremes of dependent and heterogenous observations. Statistics 50(4):930–955
Fang R, Li C, Li X (2018) Ordering results on extremes of scaled random variables with dependence and proportional hazards. Statistics 52(2):458–478
Hazra NK, Kuiti MR, Finkelstein M, Nanda AK (2017) On stochastic comparisons of maximum order statistics from the location-scale family of distributions. J Multivar Anal 160:31–41
Kayal S (2019) Stochastic comparisons of series and parallel systems with Kumaraswamy-G distributed components. Am J Math Manag Sci 38(1):1–22
Khaledi BE, Kochar SC (2002) Dispersive ordering among linear combinations of uniform random variables. J Stat Plan Inference 100(1):13–21
Khaledi BE, Farsinezhad S, Kochar SC (2011) Stochastic comparisons of order statistics in the scale model. J Stat Plan Inference 141(1):276–286
Kundu A, Chowdhury S (2017) Stochastic comparisons of lifetimes of two series and parallel systems with location-scale family distributed components having Archimedean copulas. arXiv preprint arXiv:1710.00769
Kundu A, Chowdhury S, Nanda AK, Hazra NK (2016) Some results on majorization and their applications. J Comput Appl Math 301:161–177
Li X, Fang R (2015) Ordering properties of order statistics from random variables of Archimedean copulas with applications. J Multivar Anal 133:304–320
Li C, Li X (2016) Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes. IEEE Trans Reliab 65(2):1014–1021
Li C, Li X (2019) Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables. Stat Probab Lett 146:104–111
Li C, Fang R, Li X (2016) Stochastic somparisons of order statistics from scaled and interdependent random variables. Metrika 79(5):553–578
Marshall AW, Olkin I, Arnold BC (2011) Inequality: theory of majorization and its applications. Springer series in statistics. Springer, New York
McNeil AJ, Nešlehová J (2009) Multivariate archimedean copulas, \(d\)-monotone functions and \(l^1\)-norm symmetric distributions. Ann Stat 37(5B):3059–3097
Mesfioui M, Kayid M, Izadkhah S (2017) Stochastic comparisons of order statistics from heterogeneous random variables with Archimedean copula. Metrika 80(6–8):749–766
Nadeba H, Torabi H (2018) Stochastic comparisons of series systems with independent heterogeneous Lomax-exponential components. J Stat Theory Pract 12(4):794–812
Navarro J, Spizzichino F (2010) Comparisons of series and parallel systems with components sharing the same copula. Appl Stoch Models Bus Ind 26(6):775–791
Navarro J, Torrado N, del Águila Y (2018) Comparisons between largest order statistics from multiple-outlier models with dependence. Methodol Comput Appl Probab 20(1):411–433
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, Berlin
Pledger G, Proschan F (1971) Comparisons of order statistics and of spacings from heterogeneous distributions. In: Rustagi JS (ed) Optimizing methods in statistics. Academic Press, New York, pp 89–113
Proschan F, Sethuraman J (1976) Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J Multivar Anal 6(4):608–616
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, Berlin
Singh H, Vijayasree G (1991) Preservation of partial orderings under the formation of k-out-of-n: G systems of iid components. IEEE Trans Reliab 40(3):273–276
Torrado N (2015) On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J Math Anal Appl 426(2):824–838
Torrado N (2017) Stochastic comparisons between extreme order statistics from scale models. Statistics 51(6):1359–1376
Zhang Y, Cai X, Zhao P, Wang H (2019) Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components. Statistics 53(1):126–147
Acknowledgements
The authors would like to thank the Editor in Chief Professor Maria Kateri and three anonymous reviewers for their positive remarks and useful comments. One of the authors, Sangita Das thanks the financial support provided by the MHRD, Government of India. Suchandan Kayal gratefully acknowledges the partial financial support for this research work under a Grant MTR/2018/000350, SERB, India.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have 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
About this article
Cite this article
Das, S., Kayal, S. Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples. Metrika 83, 869–893 (2020). https://doi.org/10.1007/s00184-019-00753-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-019-00753-2