Some Properties And Applications Of Shifted Proportional Stochastic Orders

Yıl 2013, Cilt: 6 Sayı: 2, 80 - 91, 01.07.2013

Jalil Jarrahiferiz Gholam Reza Mohtashami Borzadaran Abdolhamid Rezaei Roknabadi

Öz

The purpose of this paper is to study shifted versions of the proportional likelihood ratio
order, proportional (reversed) hazard rate order and some related aging classes. We give some properties and
relationships to other stochastic comparisons which are known in the literature and we study some applications
in the reliability theory. Furthermore, conditions for preservation of this orderings under weighted
distributions are given.

Anahtar Kelimeler

Up (Down) proportional likelihood ratio order, Up (Down) proportional hazard rate order, Preservation, Shock models, Coherent system

Kaynakça

• Aboukalam, F. and Kayid, M. (2007). Some new results about shifted hazard and shifted likelihood ratio orders. International Mathematical Forum, 31, 1525-1536.
• Ahmad, I. A., Kayid, M. and Li, X. (2005). The NBUT class of the life distributions. IEEE Transaction on Reliability, 396-401.
• Bartoszewicz, J. and Skolimowska, M. (2004). Stochastic ordering of weighted distributions. University of Wroclaw.
• Bartoszewicz, J. (2009). On a represervation of weighted distributions. Statistics and Probability Letters, 79, 1690-1694.
• Belzunce, F., Navarro, J., Ruiz, J. M. and Aguila, Y. D. (2004). Some results on residual entropy function. Metrika, 59, 147-161.
• Belzunce, F., Lillo, R., Ruiz, J. M. and Shaked, M. (2001). Stochastic comparisons of nonhomogeneous processes. Probability in the Engineering and Informational Sciences, 15, 199-224.
• Belzunce, F., Ruiz, J. M. and Ruiz, M. C. (2002). On preservation of some shifted and proportional orders by systems. Statistics and Probability Letters, 60, 141-154.
• Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics. Journal of Applied Probability, 31, 180-192.
• Brown, M. and Shanthikumar, J. G. (1998). Comparing the variability of random variables and point processes. Probability in the Engineering and Informational Sciences, 12, 425-444.
• Chandra, N. K. and Roy, D. (2001). Some results on reversed hazard rate. Probability in the Engineering and Informational Sciences, 15, 95-102.
• Fisher, R. A. (1934). The effects of methods of ascertainment upon the estimation of frequeneies. Annals Eugenics, 6, 13-25.
• Gupta, R. C. and Keating, J. P. (1986). Relations for reliability measures under length biased sampling. Statistical Papers, 13, 49-56.
• Gupta, R. D. and Nanda, A. K. (2001). Some results on (reversed) hazard rate ordering. Communications in Ststistics-Theory and Methods, 30, 2447-2458.
• Hu, T. and Zhu, Z. (2001). An analytic proof of the preservation of the up-shifted likelihood ratio order under convolutions. Stochastic Processes and their Applications, 95, 55-61.
• Izadkhah, S., Rezaei Roknabadi, A. H., and Mohtashami Borzadaran, G. R. (2013). On Properties of Reversed Mean Residual Life Order for Weighted Distributions. Communications in Ststistics-Theory and Methods, 42(5), 838-851.
• Jain, K., Singh, H. and Bagai, I. (1989). Relations for reliability measures of weighted distributions. Communications in Ststistics-Theory and Methods, 18, 4393-4412.
• Kayid, M. and Ahmad, I. A. (2004). On the mean inactivity time ordering with reliability applications. Probability in the Engineering and Informational Sciences, 18, 395-409.
• Kijima, M. (1998). Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains. Journal of Applied Probability, 35, 545-556.
• Kochar, S. C. and Gupta, R. P. (1987). Some results on weighted distributions for positive-valued random variables. Probability in the Engineering and Informational Sciences., 1, 417-423.
• Kochar, S. C., Li, X. and Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability, 34, 826-845.
• Lillo, R. E., Nanda, A. K. and Shaked, M. (2001). Preservation of some likelihood ratio stochastic orders by order statistics. Statistics and Probability Letters, 51, 111-119.
• Misra, N., Gupta, N. and Dhariyal, I. (2008). Preservation of some aging properties and stochastic orders by weighted distributions. Communications in Ststistics-Theory and Methods, 37, 627-644.
• Muller, A. (1997). Stochastic order generated by integrals: A unified approach. Advances in Applied Probability, 29, 414-428.
• Nakai, T. (1995). A partially observable decision problem under a shifted likelihood ratio ordering. Mathematical and Computer Modelling, 22, 237-246.
• Nanda, A. K. and Shaked, M. (2001). The hazard rate and reversed hazard rate orders with applications to order statistics. Annals of the Institute of Statistical Mathematics, Vol. 53, No. 4, 853-864.
• Patil, G. P. and Rao, C. R. (1977). The weighted distributions: A survey and their applications. Applications of Statistics, Amsterdam: North Holland, 383-405.
• Ramos-Romero, H. M. and Sordo-Diaz, M. A. (2001). The proportional likelihood ratio order and applications. Questiio, 25, 211-223.
• Rao, C. R. (1965). On discrete distributions arising out of methods of ascertainment. Pergamon Press, Oxford and Statistical Publishing Society, 320-332.
• Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Academic Press, New York.
• Shanthikumar, J. G. and Yao, D. D. (1986). The preservation of likelihood ratio ordering under convolutions. Stochastic Processes and their Applications, 23, 259-267.