In the literature on Statistical Reliability Theory and Stochastic Orders, several results based on theory of TP_{2}/RR_{2} functions have been extensively used in establishing various properties. In this paper, we provide a review of some useful results in this direction and highlight connections between them.

Type of Study: Special Issue, Original Paper |
Subject:
62Cxx: Decision theory

Received: 2020/11/21 | Accepted: 2021/02/7 | Published: 2021/06/20

Received: 2020/11/21 | Accepted: 2021/02/7 | Published: 2021/06/20

1. Aboukalam, F., and Kayid, M. (2007), Some new results about shifted hazard and shifted likelihood ratio orders. In International Mathematical Forum, 31, 1525-1536. [DOI:10.12988/imf.2007.07138]

2. Artin, E. (1931), Einfuhrung in die Theorie der Gamma-funktion. BG Teubner.

3. Barlow, R. E., and Proschan, F. (1975), Statistical theory of Reliability and life testing: Probability Models. New York: Holt, Rinehart, and Winston.

4. Bartoszewicz, J. (1998), Applications of a general composition theorem to the star order of distributions. Statistics & probability letters, 38(1),1-9. [DOI:10.1016/S0167-7152(97)00147-8]

5. Belzunce, F., Ruiz, J. M., and Ruiz, M. C. (2002), On preservation of some shifted and proportional orders by systems. Statistics & probability letters, 60(2), 141-154. [DOI:10.1016/S0167-7152(02)00302-4]

6. Belzunce, F., Martinez-Riquelme, C., and Muler, J. (2016), An Introduction to Stochastic Orders. Academic Press, Elsevier.

7. Bergmann, R. (1991), Stochastic orders and their application to a unified approach to various concepts of dependence and association. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 48-73. [DOI:10.1214/lnms/1215459849]

8. 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. [DOI:10.1017/S0269964800005301]

9. Dasgupta, S., and Sarkar, S. K. (1984), On TP2 and log-concavity. In Inequalities in Statistics and Probability, Lecture Notes-Monograph Series/ Institute of Mathematical Statistics, 54-58.

10. Dewan, I., and Khaledi, B. E. (2014), On stochastic comparisons of residual life time at random time. Statistics & Probability Letters, 88, 73-79. [DOI:10.1016/j.spl.2014.01.029]

11. Di Crescenzo, A., and Longobardi, M. (2001), The up reversed hazard rate stochastic order. Scientiae Mathematicae Japonicae, 54(3), 575-581.

12. Durham, S., Lynch, J. and Padgett, W. J. (1990), TP2-orderings and the IFR property with applications. Probability in the Engineering and Informational Sciences, 4(1), 73-88. [DOI:10.1017/S0269964800001467]

13. Hu, T., and Zhu, Z. (2001), An analytic proof of the preservation of the up-shifted likelihood ratio order under convolutions. Stochastic Process. Appl., 95, 55-61.

14. Joag-dev, K., Kochar, S., and Proschan, F. (1995), A general composition theorem and its applications to certain partial orderings of distributions. Statistics & Probabality Letters, 22,111-119. [DOI:10.1016/0167-7152(94)00056-E]

15. Karlin, S. (1968), Total positivity (Vol. 1). Stanford: Stanford University Press.

16. Keilson, J., and Sumita, U. (1982), Uniform stochastic ordering and related inequalities. Canadian Journal of Statistics, 10(3), 181-198. [DOI:10.2307/3556181]

17. Khaledi, B. E., and Shaked, M. (2010), Stochastic comparisons of multivariate mixtures. J Multivariate Anal, 101, 2486-2498. [DOI:10.1016/j.jmva.2010.06.018]

18. Khaledi, B. E. (2014), Karlin's basic composition theorems and stochastic orderings. Journal of the Iranian Statistical Society, 13(2),177-186.

19. Lai, C., and Xie, M. (2006), Stochastic ageing and dependence for reliability. Springer.

20. Laradji, A. (2015), Sums of totally positive functions of order 2 and applications. Statistics & Probability Letters, 105, 176-180. [DOI:10.1016/j.spl.2015.04.025]

21. Lehmann, E. L. (1966), Some concepts of dependence. The Annals of Mathematical Statistics, 1137-1153. [DOI:10.1214/aoms/1177699260]

22. Li, H., and Li, X. (2013), Stochastic orders in reliability and risk. In Honor of Professor Moshe Shaked. Springer.

23. Lillo, R. E., Nanda, A. K., and Shaked, M. (2000), Some shifted stochastic orders. In Recent advances in reliability theory (85-103). Birkhauser, Boston, MA.

24. Lillo, R. E., Nanda, A. K., and Shaked, M. (2001), Preservation of some likelihood ratio stochastic orders by order statistics. Statistics & Probability Letters, 51(2), 111-119. [DOI:10.1016/S0167-7152(00)00137-1]

25. Lynch, J., Mimmack, G., and Proschan, F. (1987), Uniform stochastic orderings and total positivity. Canadian Journal of Statistics, 15(1), pp.63-69. [DOI:10.2307/3314862]

26. Lynch, J. D. (1999), On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Probability in the Engineering and Informational Sciences, 13(1), 33-36. [DOI:10.1017/S0269964899131036]

27. Marshall, A. W., Olkin, I., and Arnold, B. C. (2010), Total Positivity, In: Inequalities: theory of majorization and its applications, 757-768, Springer.

28. Misra, N., and Naqvi, S. (2018), Some unified results on stochastic properties of residual lifetimes at random times. Brazilian Journal of Probability and Statistics, 32(2), 422-436. [DOI:10.1214/16-BJPS348]

29. Mosler, K., and Scarsini, M. (2012), Stochastic orders and applications: a classified bibliography (401). Springer Science & Business Media.

30. Muller, A., and Stoyan, D. (2002), Comparison methods for stochastic models and risks (389). Wiley.

31. Nakai, T. (1995), A partially observable decision problem under a shifted likelihood ratio ordering, Mathematical and Computer Modelling, 22, 237-246.

32. Nanda, A. K., Bhattacharjee, S., and Alam, S. S. (2006), On upshifted reversed mean residual life order. Communications in Statistics-Theory and Methods, 35(8), 1513-1523. [DOI:10.1080/03610920600637271]

33. Pr'ekopa, A. (1971), Logarithmic concave measures with application to stochastic programming. Acta Scientiarum Mathematicarum, 32, 301-316.

34. Pecaric, J. E., Proschan, F., and Tong, Y. L. (1992), Convex functions, partial orderings, and statistical applications, ser. Math. Sci. Engrg. Boston, MA: Academic Press, Inc, New York.

35. Shaked, M., and Shanthikumar, J. G. (2007), Stochastic Orders. New York: Springer.

36. Shanthikumar, J. G., and Yao, D. D. (1986), The preservation of likelihood ratio ordering under convolutions. Stochastic Processes and Their Applications, 23, 259-267. [DOI:10.1016/0304-4149(86)90039-6]

37. Szekli, R. (2012), Stochastic ordering and dependence in applied probability (97). Springer Science & Business Media.

38. Tong, Y. L. (2012), The multivariate normal distribution. Springer Science & Business Media.

39. Whitt, W. (1988), Stochastic orderings. Encyclopedia of the Statistical Sciences Vol. 8, S. Kotz and N.L. Johnson, editors, 832-836, Wiley, New York.

Rights and permissions | |

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |