×

Stochastic comparisons of order statistics and spacings: a review. (English) Zbl 1267.60019

Summary: We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time, we will be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters as well as the proportional hazard rate model is discussed in detail.

MSC:

60E15 Inequalities; stochastic orderings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Muller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, John Wiley & Sons, New York, NY, USA, 2002. · Zbl 0999.60002
[2] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer Series in Statistics, Springer, New York, NY, USA, 2007.
[3] H. A. David and H. N. Nagaraja, Order Statistics, John Wiley & Sons, New York, NY, USA, 3rd edition, 2003. · Zbl 1053.62060
[4] B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics, John Wiley & Sons, New York, NY, USA, 1992. · Zbl 0900.68145
[5] N. Balakrishnan and C. R. Rao, Handbook of Statistics 16: Order Statistics: Theory and Methods, Elsevier, New York, NY, USA, 1998. · Zbl 0894.00024
[6] N. Balakrishnan and C. R. Rao, Handbook of Statistics 17: Order Statistics: Applications, Elsevier, New York, NY, USA, 1998. · Zbl 0908.62060
[7] P. K. Sen, “A note on order statistics for heterogeneous distributions,” The Annals of Mathematical Statistics, vol. 41, no. 6, pp. 2137-2139, 1970. · Zbl 0216.22003 · doi:10.1214/aoms/1177696715
[8] M. Shaked and Y. L. Tong, “Stochastic ordering of spacings from dependent random variables,” Inequalities in Statistics and Probability, vol. 5, pp. 141-149, 1984.
[9] R. B. Bapat and M. I. Beg, “Order statistics for nonidentically distributed variables and permanents,” Sankhy\Ba B, vol. 51, no. 1, pp. 79-93, 1989. · Zbl 0672.62060
[10] P. J. Boland, M. Hollander, K. Joag-Dev, and S. C. Kochar, “Bivariate dependence properties of order statistics,” Journal of Multivariate Analysis, vol. 56, no. 1, pp. 75-89, 1996. · Zbl 0863.62044 · doi:10.1006/jmva.1996.0005
[11] S. C. Kochar, “Dispersive ordering of order statistics,” Statistics and Probability Letters, vol. 27, no. 3, pp. 271-274, 1996. · Zbl 0847.62039 · doi:10.1016/0167-7152(95)00083-6
[12] G. Nappo and F. Spizzichino, “Ordering properties of the TTT-plot of lifetimes with schur joint densities,” Statistics and Probability Letters, vol. 39, no. 3, pp. 195-203, 1998. · Zbl 0915.62083 · doi:10.1016/S0167-7152(98)00021-2
[13] P. J. Boland, M. Shaked, and J. G. Shanthikumar, “Stochastic ordering of order statistics,” in Handbook of Statistics 16 - Order Statistics : Theory and Methods, N. Balakrishnan and C. R. Rao, Eds., pp. 89-103, Elsevier, New York, NY, USA, 1998. · Zbl 0906.62046
[14] N. Balakrishnan, “Permanents, order statistics, outliers, and robustness,” Revista Matemática Complutense, vol. 20, no. 1, pp. 7-107, 2007. · Zbl 1148.62031 · doi:10.5209/rev_REMA.2007.v20.n1.16528
[15] P. Pledger and F. Proschan, “Comparisons of order statistics and of spacings from heterogeneous distributions,” in Optimizing Methods in Statistics, J. S. Rustagi, Ed., pp. 89-113, Academic Press, New York, NY, USA, 1971. · Zbl 0263.62062
[16] F. Proschan and J. Sethuraman, “Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability,” Journal of Multivariate Analysis, vol. 6, no. 4, pp. 608-616, 1976. · Zbl 0346.60058 · doi:10.1016/0047-259X(76)90008-7
[17] R. B. Bapat and S. C. Kochar, “On likelihood-ratio ordering of order statistics,” Linear Algebra and Its Applications, vol. 199, no. 1, pp. 281-291, 1994. · Zbl 0790.62047 · doi:10.1016/0024-3795(94)90353-0
[18] P. J. Boland, E. El-Neweihi, and F. Proschan, “Applications of the hazard rate ordering in reliability and order statistics,” Journal of Applied Probability, vol. 31, no. 1, pp. 180-192, 1994. · Zbl 0793.62053 · doi:10.2307/3215245
[19] S. C. Kochar and S. N. U. A. Kirmani, “Some results on normalized spacings from restricted families of distributions,” Journal of Statistical Planning and Inference, vol. 46, no. 1, pp. 47-57, 1995. · Zbl 0822.62081 · doi:10.1016/0378-3758(94)00095-D
[20] S. C. Kochar and R. Korwar, “Stochastic orders for spacings of heterogeneous exponential random variables,” Journal of Multivariate Analysis, vol. 57, no. 1, pp. 69-83, 1996. · Zbl 0864.62032 · doi:10.1006/jmva.1996.0022
[21] S. C. Kochar and J. Rojo, “Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions,” Journal of Multivariate Analysis, vol. 59, no. 2, pp. 272-281, 1996. · Zbl 0864.62033 · doi:10.1006/jmva.1996.0065
[22] R. Dykstra, S. C. Kochar, and J. Rojo, “Stochastic comparisons of parallel systems of heterogeneous exponential components,” Journal of Statistical Planning and Inference, vol. 65, no. 2, pp. 203-211, 1997. · Zbl 0915.62044 · doi:10.1016/S0378-3758(97)00058-X
[23] S. C. Kochar and C. Ma, “Dispersive ordering of convolutions of exponential random variables,” Statistics and Probability Letters, vol. 43, no. 3, pp. 321-324, 1999. · Zbl 0926.62005 · doi:10.1016/S0167-7152(98)00279-X
[24] J. L. Bon and E. P\valt\vanea, “Ordering properties of convolutions of exponential random variables,” Lifetime Data Analysis, vol. 5, no. 2, pp. 185-192, 1999. · Zbl 0967.60017 · doi:10.1023/A:1009605613222
[25] S. C. Kochar, “On stochastic orderings between distributions and their sample spacings,” Statistics and Probability Letters, vol. 42, no. 4, pp. 345-352, 1999. · Zbl 0956.62044 · doi:10.1016/S0167-7152(98)00224-7
[26] B.-E. Khaledi and S. C. Kochar, “Stochastic orderings between distributions and their sample spacings-II,” Statistics and Probability Letters, vol. 44, no. 2, pp. 161-166, 1999. · Zbl 1156.62331 · doi:10.1016/S0167-7152(99)00004-8
[27] B.-E. Khaledi and S. C. Kochar, “On dispersive ordering between order statistics in one-sample and two-sample problems,” Statistics and Probability Letters, vol. 46, no. 3, pp. 257-261, 2000. · Zbl 0942.62057 · doi:10.1016/S0167-7152(99)00110-8
[28] B.-E. Khaledi and S. C. Kochar, “Some new results on stochastic comparisons of parallel systems,” Journal of Applied Probability, vol. 37, no. 4, pp. 1123-1128, 2000. · Zbl 0995.62104 · doi:10.1239/jap/1014843091
[29] B. Khaledi and S. C. Kochar, “Sample range-some stochastic comparisons results,” Calcutta Statistical Association Bulletin, vol. 50, pp. 283-291, 2000. · Zbl 0994.60014
[30] B.-E. Khaledi and S. C. Kochar, “Stochastic properties of spacings in a single-outlier exponential model,” Probability in the Engineering and Informational Sciences, vol. 15, no. 3, pp. 401-408, 2001. · Zbl 0987.60031 · doi:10.1017/S0269964801153088
[31] S. M. Ross, Stochastic Processes, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0555.60002
[32] J. G. Shanthikumar and D. D. Yao, “Bivariate characterization of some stochastic order relations,” Advances in Applied Probability, vol. 23, pp. 642-659, 1991. · Zbl 0745.62054 · doi:10.2307/1427627
[33] M. Hollander, F. Proschan, and J. Sethuraman, “Functions decreasing in transposition and their applications in ranking problems,” The Annals of Statistics, vol. 5, no. 4, pp. 722-733, 1977. · Zbl 0356.62043 · doi:10.1214/aos/1176343895
[34] A. W. Marshall, I. Olkin, and B. Arnold, Inequalities : Theory of Majorization and Its Applications, Springe, New York, NY, USA, 2nd edition, 2010. · Zbl 1219.26003
[35] T. Robertson and F. T. Wright, “On measuring the conformity of a parameter set to a trend, with applications,” Annals of Statistics, vol. 10, no. 4, pp. 1234-1245, 1982. · Zbl 0512.62033 · doi:10.1214/aos/1176345988
[36] K. Doksum, “Star shaped transformations and the power of rank tests,” Annals of Mathematical Statistics, vol. 40, no. 4, pp. 1167-1176, 1969. · Zbl 0188.50601 · doi:10.1214/aoms/1177697493
[37] J. V. Deshpande and S. C. Kochar, “Dispersive ordering is the same as tail ordering,” Advances in Appllied Probability, vol. 15, pp. 686-687, 1983. · Zbl 0516.60015 · doi:10.2307/1426626
[38] D. J. Saunders, “Dispersive ordering of distributions,” Advances in Applied Probability, vol. 16, pp. 693-694, 1984. · Zbl 0547.60023 · doi:10.2307/1427297
[39] I. W. Saunders and P. A. P. Moran, “On quantiles of the gamma and F distributions,” Journal of Applied Probability, vol. 15, no. 2, pp. 426-432, 1978. · Zbl 0401.62015 · doi:10.2307/3213414
[40] T. Lewis and J. W. Thompson, “Dispersive distribution and the connection between dispersivity and strong unimodality,” Journal of Applied Probability, vol. 18, no. 1, pp. 76-90, 1981. · Zbl 0453.60023 · doi:10.2307/3213168
[41] I. Bagai and S. C. Kochar, “On tail-ordering and comparison of failure rates,” Communications in Statistics, vol. 15, no. 4, pp. 1377-1388, 1986. · Zbl 0595.62041 · doi:10.1080/03610928608829189
[42] J. Bartoszewicz, “Dispersive ordering and the total time on test transformation,” Statistics and Probability Letters, vol. 4, no. 6, pp. 285-288, 1986. · Zbl 0622.62052 · doi:10.1016/0167-7152(86)90045-3
[43] J. Bartoszewicz, “A note on dispersive ordering defined by hazard functions,” Statistics and Probability Letters, vol. 6, no. 1, pp. 13-16, 1987. · Zbl 0633.62017 · doi:10.1016/0167-7152(87)90052-6
[44] S. C. Kochar, “Distribution-free comparison of two probability distributions with reference to their hazard rates,” Biometrika, vol. 66, no. 3, pp. 437-441, 1979. · Zbl 0426.62027 · doi:10.1093/biomet/66.3.437
[45] J. Jeon, S. C. Kochar, and C. G. Park, “Dispersive ordering-some applications and examples,” Statistical Papers, vol. 47, no. 2, pp. 227-247, 2006. · Zbl 1105.62014 · doi:10.1007/s00362-005-0285-4
[46] J. M. Fernandez-Ponce, S. C. Kochar, and J. Muñoz-Perez, “Partial orderings of distributions based on right-spread functions,” Journal of Applied Probability, vol. 35, no. 1, pp. 221-228, 1998. · Zbl 0898.62124 · doi:10.1239/jap/1032192565
[47] M. Shaked and J. G. Shanthikumar, “Two variability orders,” Probability in the Engineering and Informational Sciences, vol. 12, no. 1, pp. 1-23, 1998. · Zbl 0984.60023 · doi:10.1017/S0269964800005039
[48] W. R. van Zwet, Convex Transformations of Random Variables, vol. 7 of Mathematical Centre Tracts, Mathematical Centre, Amsterdam, The Netherlands, 2nd edition, 1964. · Zbl 0125.37102
[49] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, Silver Spring, Maryland, MD, USA, 1981. · Zbl 0379.62080
[50] S. C. Kochar, “On extensions of DMRL and related partial orderings of life distributions,” Stochastic Models, vol. 5, no. 2, pp. 235-246, 1989. · Zbl 0674.62063 · doi:10.1080/15326348908807107
[51] E. L. Lehmann, “Some concepts of dependence,” Annals of Mathematical Statistics, vol. 37, no. 5, pp. 1137-1153, 1966. · Zbl 0146.40601 · doi:10.1214/aoms/1177699260
[52] J. H. B. Kemperman, “On the FKG-inequality for measures on a partially ordered space,” Indagationes Mathematicae, vol. 39, no. 4, pp. 313-331, 1977. · Zbl 0384.28012
[53] H. W. Block and M. L. Ting, “Some concepts of multivariate dependence,” Communications in Statistics, vol. 10, no. 8, pp. 749-762, 1981. · Zbl 0478.62045 · doi:10.1080/03610928108828072
[54] S. Karlin and Y. Rinott, “Classes of orderings of measures and related correlation inequalities. I. multivariate totally positive distributions,” Journal of Multivariate Analysis, vol. 10, no. 4, pp. 467-498, 1980. · Zbl 0469.60006 · doi:10.1016/0047-259X(80)90065-2
[55] H. Joe, Multivariate Models and Dependence Concepts, Chapman & Hall, London, UK, 1997. · Zbl 0990.62517
[56] R. B. Nelsen, “On measures of association as measures of positive dependence,” Statistics and Probability Letters, vol. 14, no. 4, pp. 269-274, 1992. · Zbl 0761.62075 · doi:10.1016/0167-7152(92)90056-B
[57] J. Avérous, C. Genest, and S. C. Kochar, “On the dependence structure of order statistics,” Journal of Multivariate Analysis, vol. 94, no. 1, pp. 159-171, 2005. · Zbl 1065.62087 · doi:10.1016/j.jmva.2004.03.004
[58] A. Dolati, C. Genest, and S. C. Kochar, “On the dependence between the extreme order statistics in the proportional hazards model,” Journal of Multivariate Analysis, vol. 99, no. 5, pp. 777-786, 2008. · Zbl 1136.60011 · doi:10.1016/j.jmva.2007.03.001
[59] R. B. Nelsen, An Introduction to Copulas, Lecture Notes in Statistics, Springer, New York, NY, USA, 2nd edition, 2006. · Zbl 1152.62030
[60] C. Genest, S. C. Kochar, and M. Xu, “On the range of heterogeneous samples,” Journal of Multivariate Analysis, vol. 100, no. 8, pp. 1587-1592, 2009. · Zbl 1166.60015 · doi:10.1016/j.jmva.2009.01.001
[61] M. Z. Raqab and W. A. Amin, “Some ordering results on order statistics and record values,” IAPQR Transactions, vol. 21, pp. 1-8, 1996. · Zbl 0899.62070
[62] H. A. David and R. A. Groeneveld, “Measures of local variation in a distribution: expected length of spacings and variances of order statistics,” Biometrika, vol. 69, no. 1, pp. 227-232, 1982. · Zbl 0488.62035 · doi:10.1093/biomet/69.1.227
[63] B. C. Arnold and J. A. Villasenor, “Lorenz ordering of order statistics,” in Stochastic Orders and Decision under Risk, vol. 19 of IMS Lecture Notes Monograph Series, pp. 38-47, Institute of Mathematical Statistics, Hayward, Calif, USA, 1991. · Zbl 0755.62045 · doi:10.1214/lnms/1215459848
[64] B. C. Arnold and H. N. Nagaraja, “Lorenz ordering of exponential order statistics,” Statistics and Probability Letters, vol. 11, no. 6, pp. 485-490, 1991. · Zbl 0724.62052 · doi:10.1016/0167-7152(91)90112-5
[65] B. Wilfling, “Lorenz ordering of power-function order statistics,” Statistics and Probability Letters, vol. 30, no. 4, pp. 313-319, 1996. · Zbl 0885.62061 · doi:10.1016/S0167-7152(95)00234-0
[66] C. Kleiber, “Variability ordering of heavy-tailed distributions with applications to order statistics,” Statistics and Probability Letters, vol. 58, no. 4, pp. 381-388, 2002. · Zbl 1014.62058 · doi:10.1016/S0167-7152(02)00151-7
[67] S. C. Kochar, “Lorenz ordering of order statistics,” Statistics and Probability Letters, vol. 76, no. 17, pp. 1855-1860, 2006. · Zbl 1102.60016 · doi:10.1016/j.spl.2006.04.032
[68] P. J. Bickel, “Some contributions to the theory of order statistics,” in Fifth Berkeley Symposium on Mathematics and Statistics, L. M. LeCam and J. Neyman, Eds., vol. 1, pp. 575-591, University of California Press, Berkeley, Calif, USA, 1967. · Zbl 0214.46602
[69] J. W. Tukey, “A problem of berkson, and minimum variance orderly estimators,” The Annals of Mathematical Statististis, vol. 29, no. 2, pp. 588-592, 1958. · Zbl 0086.35601 · doi:10.1214/aoms/1177706637
[70] S. H. Kim and H. A. David, “On the dependence structure of order statistics and concomitants of order statistics,” Journal of Statistical Planning and Inference, vol. 24, no. 3, pp. 363-368, 1990. · Zbl 0698.62050 · doi:10.1016/0378-3758(90)90055-Y
[71] M. Scarsini, “On measures of concordance,” Stochastica, vol. 8, no. 3, pp. 201-218, 1984. · Zbl 0582.62047
[72] G. A. Fredricks and R. B. Nelsen, “On the relationship between Spearman’s rho and Kendall’s tau for pairs of continuous random variables,” Journal of Statistical Planning and Inference, vol. 137, no. 7, pp. 2143-2150, 2007. · Zbl 1120.62045 · doi:10.1016/j.jspi.2006.06.045
[73] V. Schmitz, “Revealing the dependence structure between x(1) and x(n),” Journal of Statistical Planning and Inference, vol. 123, no. 1, pp. 41-47, 2004. · Zbl 1095.62073 · doi:10.1016/S0378-3758(03)00143-5
[74] X. Li and Z. Li, “Proof of a conjecture on Spearman’s \rho and Kendall’s \tau for sample minimum and maximum,” Journal of Statistical Planning and Inference, vol. 137, no. 1, pp. 359-361, 2007. · Zbl 1126.62047 · doi:10.1016/j.jspi.2005.08.048
[75] Y. P. Chen, “A note on the relationship between Spearman’s \rho and Kendall’s \tau for extreme order statistics,” Journal of Statistical Planning and Inference, vol. 137, no. 7, pp. 2165-2171, 2007. · Zbl 1120.62031 · doi:10.1016/j.jspi.2006.07.003
[76] J. Navarro and N. Balakrishnan, “Study of some measures of dependence between order statistics and systems,” Journal of Multivariate Analysis, vol. 101, no. 1, pp. 52-67, 2010. · Zbl 1183.62099 · doi:10.1016/j.jmva.2009.04.016
[77] T. Hu and C. Xie, “Negative dependence in the balls and bins experiment with applications to order statistics,” Journal of Multivariate Analysis, vol. 97, no. 6, pp. 1342-1354, 2006. · Zbl 1103.62042 · doi:10.1016/j.jmva.2005.09.008
[78] D. Dubhashi and O. Häggström, “A note on conditioning and stochastic domination for order statistics,” Journal of Applied Probability, vol. 45, no. 2, pp. 575-579, 2008. · Zbl 1141.62034 · doi:10.1239/jap/1214950369
[79] T. Hu and H. Chen, “Dependence properties of order statistics,” Journal of Statistical Planning and Inference, vol. 138, no. 7, pp. 2214-2222, 2008. · Zbl 1138.60024 · doi:10.1016/j.jspi.2007.09.013
[80] W. Zhuang, J. Yao, and T. Hu, “Conditional ordering of order statistics,” Journal of Multivariate Analysis, vol. 101, no. 3, pp. 640-644, 2010. · Zbl 1185.60018 · doi:10.1016/j.jmva.2009.11.007
[81] J. Lynch, G. Mimmack, and F. Proschan, “Uniform stochastic orderings and total positivity,” The Canadian Journal of Statistics, vol. 15, no. 1, pp. 63-69, 1987. · Zbl 0617.62005 · doi:10.2307/3314862
[82] W. Chan, F. Proschan, and J. Sethuraman, “Convex-ordering among functions, with applications to reliability and mathematical statistics,” in Topics in Statistical Dependence, vol. 16 of IMS Lecture Notes Monograph Series, pp. 121-134, Institute of Mathematical Statistics, Hayward, Calif, USA, 1990. · Zbl 0770.62086 · doi:10.1214/lnms/1215457555
[83] R. E. Lillo, A. K. Nanda, and M. Shaked, “Preservation of some likelihood ratio stochastic orders by order statistics,” Statistics and Probability Letters, vol. 51, no. 2, pp. 111-119, 2001. · Zbl 0982.60009 · doi:10.1016/S0167-7152(00)00137-1
[84] N. Torrado and J. J. P. Veerman, “Asymptotic reliability theory of k-out-of-n systems,” Journal of Statistical Planning and Inference, vol. 142, no. 9, pp. 2646-2665, 2012. · Zbl 1260.62079
[85] S. C. Kochar and M. Xu, “Stochastic comparisons of parallel systems when components have proportional hazard rates,” Probability in the Engineering and Informational Sciences, vol. 21, no. 4, pp. 597-609, 2007. · Zbl 1142.62084 · doi:10.1017/S0269964807000344
[86] J. L. Bon and E. P\valt\vanea, “Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables,” ESAIM-Probability and Statistics, vol. 10, pp. 1-10, 2006. · Zbl 1186.90042 · doi:10.1051/ps:2005020
[87] S. C. Kochar and M. Xu, “Comparisons of parallel systems according to the convex transform order,” Journal of Applied Probability, vol. 46, no. 2, pp. 342-352, 2009. · Zbl 1168.60318 · doi:10.1239/jap/1245676091
[88] S. C. Kochar and M. Xu, “On the skewness of order statistics with applications,” Annals of Operations Research. In press. · Zbl 1291.62100
[89] E. P\valt\vanea, “On the comparison in hazard rate ordering of fail-safe systems,” Journal of Statistical Planning and Inference, vol. 138, no. 7, pp. 1993-1997, 2008. · Zbl 1138.60025 · doi:10.1016/j.jspi.2007.08.001
[90] P. Zhao and N. Balakrishnan, “Characterization of MRL order of fail-safe systems with heterogeneous exponential components,” Journal of Statistical Planning and Inference, vol. 139, no. 9, pp. 3027-3037, 2009. · Zbl 1181.62171 · doi:10.1016/j.jspi.2009.02.006
[91] P. Zhao, X. Li, and N. Balakrishnan, “Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables,” Journal of Multivariate Analysis, vol. 100, no. 5, pp. 952-962, 2009. · Zbl 1167.62048 · doi:10.1016/j.jmva.2008.09.010
[92] P. Zhao and N. Balakrishnan, “Dispersive ordering of fail-safe systems with heterogeneous exponential components,” Metrika, vol. 74, no. 2, pp. 203-210, 2011. · Zbl 1434.62218 · doi:10.1007/s00184-010-0297-5
[93] T. Hu, “Monotone coupling and stochastic ordering of order statistics,” Journal of Systems Science and Complexity, vol. 8, no. 3, pp. 209-214, 1995. · Zbl 0852.62052
[94] L. Sun and X. Zhang, “Stochastic comparisons of order statistics from gamma distributions,” Journal of Multivariate Analysis, vol. 93, no. 1, pp. 112-121, 2005. · Zbl 1058.60014 · doi:10.1016/j.jmva.2004.01.009
[95] B.-E. Khaledi and S. C. Kochar, “Weibull distribution: some stochastic comparisons results,” Journal of Statistical Planning and Inference, vol. 136, no. 9, pp. 3121-3129, 2006. · Zbl 1095.62019 · doi:10.1016/j.jspi.2004.12.013
[96] E. W. Stacy, “A generalization of the gamma distribution,” Annals of Mathematical Statistics, vol. 33, no. 3, pp. 1187-1192, 1962. · Zbl 0121.36802 · doi:10.1214/aoms/1177704481
[97] B.-E. Khaledi, S. Farsinezhad, and S. C Kochar, “Stochastic comparisons of order statistics in the scale model,” Journal of Statistical Planning and Inference, vol. 141, no. 1, pp. 276-286, 2011. · Zbl 1207.62108 · doi:10.1016/j.jspi.2010.06.006
[98] V. Bagdonavicius and M. Nikulin, Accelerated Life Models, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. · Zbl 1100.62544
[99] N. Balakrishnan and P. Zhao, “Hazard rate comparison of parallel systems with heterogeneous gamma components,” Journal of Multivariate Analysis, vol. 113, pp. 153-160. · Zbl 1253.60022 · doi:10.1016/j.jmva.2011.05.001
[100] Y. S. Sathe, “On the correlation coefficient between the first and the r-th smallest order statistics based on n independent exponential random variables,” Communications in Statistics, vol. 17, pp. 3295-3299, 1988. · Zbl 0696.62222 · doi:10.1080/03610928808829804
[101] S. R. Jammalamadaka and E. Taufer, “Testing exponentiality by comparing the empirical distribution function of the normalized spacings with that of the original data,” Journal of Nonparametric Statistics, vol. 15, no. 6, pp. 719-729, 2003. · Zbl 1091.62513 · doi:10.1080/10485250310001638102
[102] S. R. Jammalamadaka and M. N. Goria, “A test of goodness-of-fit based on Gini’s index of spacings,” Statistics and Probability Letters, vol. 68, no. 2, pp. 177-187, 2004. · Zbl 1058.62039 · doi:10.1016/j.spl.2004.02.009
[103] M. Xu and X. Li, “Some further results on the winner’s rent in the secondprice business auction,” Sankhya, vol. 70, no. 1, pp. 124-133, 2008. · Zbl 1192.91110
[104] R. E. Barlow and F. Proschan, “Inequalities for linear combinations of order statistics from restricted families,” Annals of Mathematical Statistics, vol. 37, pp. 1574-1592, 1966. · Zbl 0149.15402 · doi:10.1214/aoms/1177699149
[105] N. Misra and E. C. van der Meulen, “On stochastic properties of m-spacings,” Journal of Statistical Planning and Inference, vol. 115, no. 2, pp. 683-697, 2003. · Zbl 1016.62060 · doi:10.1016/S0378-3758(02)00157-X
[106] N. Torrado, R. E. Lillo, and M. P. Wiper, “On the conjecture of Kochar and Korwar,” Journal of Multivariate Analysis, vol. 101, no. 5, pp. 1274-1283, 2010. · Zbl 1192.60051 · doi:10.1016/j.jmva.2009.11.006
[107] S. Wen, Q. Lu, and T. Hu, “Likelihood ratio orderings of spacings of heterogeneous exponential random variables,” Journal of Multivariate Analysis, vol. 98, no. 4, pp. 743-756, 2007. · Zbl 1115.60022 · doi:10.1016/j.jmva.2006.08.011
[108] M. Xu, X. Li, P. Zhao, and Z. Li, “Likelihood ratio order of m-spacings in multiple-outlier models,” Communications in Statistics, vol. 36, no. 8, pp. 1507-1525, 2007. · Zbl 1119.60016 · doi:10.1080/03610920601125839
[109] H. Chen and T. Hu, “Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables,” Metrika, vol. 68, no. 1, pp. 17-29, 2008. · Zbl 1433.62122 · doi:10.1007/s00184-007-0140-9
[110] S. C. Kochar, X. Li, and M. Xu, “Excess wealth order and sample spacings,” Statistical Methodology, vol. 4, no. 4, pp. 385-392, 2007. · Zbl 1248.91043 · doi:10.1016/j.stamet.2006.11.002
[111] F. Belzunce, “On a characterization of right spread order by the increasing convex order,” Statistics and Probability Letters, vol. 45, no. 2, pp. 103-110, 1999. · Zbl 0941.60039 · doi:10.1016/S0167-7152(99)00048-6
[112] S. C. Kochar and M. Xu, “Stochastic comparisons of spacings from heterogeneous samples,” in Advanced in Directional and Linear Statistics: A Festschrift for Sreenivasa Rao Jammalamadaka, M. Wells and A. Sengupta, Eds., pp. 113-1129, Springer, New York, NY, USA, 2011.
[113] N. Balakrishnan and M. Xu, “On the sample ranges from heterogeneous exponential variables,” Journal of Multivariate Analysis, vol. 109, pp. 1-9, 2012. · Zbl 1247.60028 · doi:10.1016/j.jmva.2012.02.009
[114] N. Torrado and R. E. Lillo, “Likelihood ratio order of spacings from two heterogeneous samples,” Journal of Multivariate Analysis, vol. 114, pp. 338-348, 2013. · Zbl 1255.62148
[115] W. Ding, G. Da, and P. Zhao, “On sample ranges from two heterogeneous random variables,” Journal of Multivariate Analysis. · Zbl 1278.90113
[116] B.-E. Khaledi and S. C. Kochar, “Dependence among spacings,” Probability in the Engineering and Informational Sciences, vol. 14, no. 4, pp. 461-472, 2000. · Zbl 0971.60020 · doi:10.1017/S0269964800144055
[117] B. Epstein, “Estimation from life test data,” Technometric, vol. 2, no. 4, pp. 447-454, 1960. · Zbl 0096.12004 · doi:10.2307/1266453
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.