Abstract
Let \(X_{1},\ldots , X_{n}\) be lifetimes of components with independent non-negative generalized Birnbaum–Saunders random variables with shape parameters \(\alpha _{i}\) and scale parameters \(\beta _{i},~ i=1,\ldots ,n\), and \(I_{p_{1}},\ldots , I_{p_{n}}\) be independent Bernoulli random variables, independent of \(X_{i}\)’s, with \(E(I_{p_{i}})=p_{i},~i=1,\ldots ,n\). These are associated with random shocks on \(X_{i}\)’s. Then, \(Y_{i}=I_{p_{i}}X_{i}, ~i=1,\ldots ,n,\) correspond to the lifetimes when the random shock does not impact the components and zero when it does. In this paper, we discuss stochastic comparisons of the smallest order statistic arising from such random variables \(Y_{i},~i=1,\ldots ,n\). When the matrix of parameters \((h({\varvec{p}}), {\varvec{\beta }}^{\frac{1}{\nu }})\) or \((h({\varvec{p}}), {\varvec{\frac{1}{\alpha }}})\) changes to another matrix of parameters in a certain mathematical sense, we study the usual stochastic order of the smallest order statistic in such a setup. Finally, we apply the established results to two special cases: classical Birnbaum–Saunders and logistic Birnbaum–Saunders distributions.
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References
Balakrishnan N, Haidari A, Masoumifard K (2015) Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Trans Reliab 64:333–348
Balakrishnan N, Zhao P (2013) Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Prob Eng Inf Sci 27:403–469 (with discussions)
Birnbaum ZW, Saunders SC (1969a) A new family of life distributions. J Appl Probab 6:319–327
Birnbaum ZW, Saunders SC (1969b) Estimation for a family of life distributions with applications to fatigue. J Appl Probab 6:328–347
Chang DS, Tang LC (1993) Reliability bounds and critical time for the Birnbaum–Saunders distribution. IEEE Trans Reliab 47:88–95
Diáz-Garciá JA, Leiva-Sánchez V (2005) A new family of life distributions based on the elliptically contoured distributions. J Stat Plan Inference 128:445–457
Dupuis DJ, Mills JE (1994) Robust estimation of the Birnbaum–Saunders distribution. IEEE Trans Reliab 42:464–469
Dykstra R, Kochar SC, Rojo J (1997) Stochastic comparisons of parallel systems of heterogeneous exponential components. J Stat Plan Inference 65:203–211
Fang L, Zhu X, Balakrishnan N (2016) Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum–Saunders components. Stat Probab Lett 112:131–136
Fang L, Zhang X (2010) Slepian’s inequality with respect to majorization. Linear Algebra Appl 434:1107–1118
Khaledi B, Farsinezhad S, Kochar SC (2011) Stochastic comparisons of order statistics in the scale models. J Stat Plan Inference 141:276–286
Khaledi B, Kochar SC (2000) Some new results on stochastic comparisons of parallel systems. J Appl Probab 37:1123–1128
Kundu D, Kannan N, Balakrishnan N (2008) On the hazard function of Birnbaum–Saunders distribution and associated inference. Comput Stat Data Anal 52:2692–2702
Kundu D, Balakrishnan N, Jamalizadeh A (2013) Generalized multivariate Birnbaum–Saunders distributions and related inferential issues. J Multivar Anal 116:230–244
Leiva V, Riquelme M, Balakrishnan N, Sanhueza A (2008a) Lifetime analysis based on the generalized Birnbaum–Saunders distribution. Comput Stat Data Anal 21:2079–2097
Leiva V, Barros M, Paula GA, Sanhueza A (2008b) Generalized Birnbaum–Saunders distribution applied to air pollutant concentration. Environmetrics 19:235–249
Leiva V, Sanhueza A, Sen PK, Paula GA (2008c) Random number generators for the generalized Birnbaum–Saunders distribution. J Stat Comput Simul 78:1105–1118
Li C, Li X (2015) Likelihood ratio order of sample minimum from heterogeneous Weibull random variables. Stat Probab Lett 97:46–53
Ng HKT, Kundu D, Balakrishnan N (2003) Modified moment estimation for the two-parameter Birnbaum–Saunders distribution. Comput Stat Data Anal 43:283–298
Ng HKT, Kundu D, Balakrishnan N (2006) Point and interval estimations for the two-parameter Birnbaum–Saunders distribution based on Type-II censored samples. Comput Stat Data Anal 50:3222–3242
Sanhueza A, Leiva V, Balakrishnan N (2008) The generalized Birnbaum–Saunders distribution and its theory, methodology, and application. Commun Stat Theory Methods 37:645–670
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York
Zhu X, Balakrishnan N (2015) Birnbaum–Saunders distribution based on Laplace kernel and some properties and inferential issues. Stat Probab Lett 101:1–10
Zhao P, Zhang Y (2014) On the maxima of heterogeneous gamma variables with different shape and scale parameters. Metrika 77:811–836
Zhao P, Hu Y, Zhang Y (2015) Some new results on largest order statistics from multiple-outlier gamma models. Probab Eng Inf Sci 29:597–621
Acknowledgements
This research was supported by the Provincial Natural Science Research Project of Anhui Colleges (Nos. KJ2016A263, KJ2017ZD27), the National Natural Science Foundation of Anhui Province (No. 1608085J06), and the PhD research startup foundation of Anhui Normal University (No. 2014bsqdjj34). The research work of the last author was supported by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
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Fang, L., Balakrishnan, N. Ordering properties of the smallest order statistics from generalized Birnbaum–Saunders models with associated random shocks. Metrika 81, 19–35 (2018). https://doi.org/10.1007/s00184-017-0632-1
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DOI: https://doi.org/10.1007/s00184-017-0632-1