Abstract
In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.
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Belzunce, F., Franco, M., Ruiz, J. M. and Ruiz, M. C. (2001), On partial orderings between coherent systems with different structure, Probab. Engrg. Inform. Sci., 15, 273-293.
Block, H. W., Savits, T. H. and Singh, H. (1998). The reversed hazard rate function, Probab. Engrg. Inform. Sci., 12, 69-90.
Boland, P. J. and Proschan, F. (1994). Stochastic order in system reliability theory, Stochastic Orders and Their Applications (eds. M. Shaked and J. G. Shanthikumar), 485-508, Academic Press, San Diego.
Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics, J. Appl. Probab., 31, 180-192.
Boland, P. J., Hu, T., Shaked, M. and Shanthikumar, J. G. (2001). Stochastic ordering of order statistics, Essays on Uncertainty (eds. M. Dror, P. L'Ecuyer and F. Szidarovsky) (to appear).
Cheng, D. W. and Righter, R. (1995). On the order of tandem queues, Queueing Systems, 21, 143-160.
Cheng, D. W. and Zhu, Y. (1993). Optimal order of servers in a tandem queue with general blocking, Queueing Systems, 14, 427-437.
Eeckhoudt, L. and Gollier, C. (1995). Demand for risky assets and the monotone probability ratio order, Journal of Risk and Uncertainty, 11, 113-122.
Hu, T. and He, F. (2000). A note on comparisons of k-out-of-n systems with respect to the hazard and reversed hazard rate orders, Probab. Engrg. Inform. Sci., 14, 27-32.
Khaledi, B.-H. and Kochar, S. (1999). Stochastic orderings between distributions and their sample spacings—II, Statist. Probab. Lett., 44, 161-166.
Kijima, M. (1998). Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains, J. Appl. Probab., 35, 545-556.
Kijima, M. and Ohnishi, M. (1999). Stochastic orders and their applications in financial optimization, Math. Methods Oper. Res., 50, 351-372.
Kochar, S. C. and Kirmani, S. N. U. A. (1995). Some results on normalized spacings from restricted families of distributions, J. Statist. Plann. Inference, 46, 47-57.
Lillo, R. E., Nanda, A. K. and Shaked, M. (2001). Preservation of some likelihood ratio stochastic orders by order statistics, Statist. Probab. Lett., 51, 111-119.
Ma, C. (1999). Uniform stochastic ordering on a system of components with dependent lifetimes induced by a common environment, Sankhyā Ser. A, 61, 218-228.
Nanda, A. K., Jain, K. and Singh, H. (1998). Preservation of some partial orderings under the formation of coherent systems, Statist. Probab. Lett., 39, 123-131.
Sengupta, D., Singh, H. and Nanda, A. K. (1999). The proportional reversed hazards model, Tech. Report, Applied Statistics Division, Indian Statistical Institute, Calcutta.
Shaked, M. and Shanthikumar, J. G. (1988). On the first-passage times of pure jump processes, J. Appl. Probab., 25, 501-509.
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications, Academic Press, San Diego.
Shaked, M. and Wong, T. (1997a). Stochastic orders based on ratios of Laplace transforms, J. Appl. Probab., 34, 404-419.
Shaked, M. and Wong, T. (1997b). Stochastic comparisons of random minima and maxima, J. Appl. Probab., 34, 420-425.
Shaked, M., Shanthikumar, J. G. and Tong, Y. L. (1995). Parametric Schur convexity and arrangement monotonicity properties of partial sums, J. Multivariate Anal., 53, 293-310.
Shanthikumar, J. G., Yamazaki, G. and Sakasegawa, H. (1991). Characterization of optimal order of servers in a tandem queue with blocking, Oper. Res. Lett., 10, 17-22.
Singh, H. and Vijayasree, G. (1991). Preservation of partial orderings under the formation of k-out-of-n:G systems of i.i.d. components, IEEE Transactions on Reliability, 40, 273-276.
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Nanda, A.K., Shaked, M. The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics. Annals of the Institute of Statistical Mathematics 53, 853–864 (2001). https://doi.org/10.1023/A:1014677608075
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DOI: https://doi.org/10.1023/A:1014677608075