Applications of Mathematics, online first, 15 pp.

Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran

Received June 20, 2016.   First published August 16, 2017.

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran, Department of Statistics, Ordered and Spatial Data Center of Excellence Faculty of Mathematical Sciences Ferdowsi University of Mashhad, Mashhad, Iran, e-mail:,,

Abstract: Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser's function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via $w(\cdot)$-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable $X$ is provided.

Keywords: characterization; hazard rate; mean residual life function; reversed hazard rate; expected inactivity time; log-odds rate; Glaser's function

Classification (MSC 2010): 60E15, 62E10

DOI: 10.21136/AM.2017.0182-16

Full text available as PDF.

[1] M. Asadi, A. Berred: Properties and estimation of the mean past lifetime. Statistics 46 (2012), 405-417. DOI 10.1080/02331888.2010.540666 | MR 2929163 | Zbl 1241.62140
[2] M. Asadi, Y. Zohrevand: On the dynamic cumulative residual entropy. J. Stat. Plann. Inference 137 (2007), 1931-1941. DOI 10.1016/j.jspi.2006.06.035 | MR 2323874 | Zbl 1118.62006
[3] M. C. Bhattacharjee: The class of mean residual lives and some consequences. SIAM J. Algebraic Discrete Methods 3 (1982), 56-65. DOI 10.1137/0603006 | MR 0644957 | Zbl 0495.60091
[4] S. Bhattacharjee, A. K. Nanda, S. K. Misra: Inequalities involving expectations to characterize distributions. Stat. Probab. Lett. 83 (2013), 2113-2118. DOI 10.1016/j.spl.2013.05.022 | MR 3079054 | Zbl 1285.60010
[5] T. Cacoullos, V. Papathanasiou: On upper bounds for the variance of functions of random variables. Stat. Probab. Lett. 3 (1985), 175-184. DOI 10.1016/0167-7152(85)90014-8 | MR 0801687 | Zbl 0572.60021
[6] T. Cacoullos, V. Papathanasiou: Characterizations of distributions by variance bounds. Stat. Probab. Lett. 7 (1989), 351-356. DOI 10.1016/0167-7152(89)90050-3 | MR 1001133 | Zbl 0677.62012
[7] N. K. Chandra, D. Roy: Some results on reversed hazard rate. Probab. Eng. Inf. Sci. 15 (2001), 95-102. DOI 10.1017/S0269964801151077 | MR 1825537 | Zbl 1087.62510
[8] M. Fradelizi, M. Madiman, L. Wang: Optimal concentration of information content for log-concave densities. High Dimensional Probability VII (C. Houdré, et al., eds.). Progress in Probability 71, Birkhäuser, Springer (2016), pp. 45-60. DOI 10.1007/978-3-319-40519-3_3 | MR 3565259 | Zbl 1358.60036
[9] F. Guess, F. Proschan: Mean residual life: Theory and applications. Handbook of Statistics 7 (P. R. Krishnaiah, et al., eds.). Elsevier Science Publishers, Atlanta (1988), pp. 215-224. DOI 10.1016/S0169-7161(88)07014-2
[10] R. C. Gupta: On the mean residual life function in survival studies. Statistical Distributions in Scientific Work, Vol. 5 (Trieste, 1980) Proc. NATO Adv. Study Inst. (Trieste 1980), Reidel, Dordrecht (1981), 327-334. DOI 10.1007/978-94-009-8552-0_26 | MR 0656346 | Zbl 0474.62092
[11] R. C. Gupta, S. N. U. A. Kirmani: Some characterization of distributions by functions of failure rate and mean residual life. Commun. Stat., Theory Methods 33 (2004), 3115-3131. DOI 10.1081/STA-200039060 | MR 2138677 | Zbl 1087.62014
[12] R. C. Gupta, R. Warren: Determination of change points of non-monotonic failure rates. Commun. Stat., Theory Methods 30 (2001), 1903-1920. DOI 10.1081/STA-100105704 | MR 1861623 | Zbl 0991.62085
[13] A. Gut: Probability: A Graduate Course. Springer Texts in Statistics, Springer, New York (2013). DOI 10.1007/978-1-4614-4708-5 | MR 2977961 | Zbl 1267.60001
[14] W. J. Hall, J. A. Wellner: Mean residual life. Statistics and Related Topics (M. Csörgö, et al., eds.). Proc. Int. Symp. (Ottawa 1980), North Holland Publishing, Amsterdam (1981), pp. 169-184. MR 0665274 | Zbl 0481.62078
[15] M. Iwińska, M. Szymkowiak: Characterizations of distributions through selected functions of reliability theory. Commun. Stat., Theory Methods 46 (2017), 69-74. DOI 10.1080/03610926.2014.985837 | MR 3553015 | Zbl 06708605
[16] R. Jiang, P. Ji, X. Xiao: Aging property of unimodal failure rate models. Reliab. Eng. Syst. Saf. 79 (2003), 113-116. DOI 10.1016/S0951-8320(02)00175-8
[17] O. Johnson: Information Theory and the Central Limit Theorem. Imperial College Press, London (2004). MR 2109042 | Zbl 1061.60019
[18] C. Kundu, A. Ghosh: Inequalities involving expectations of selected functions in reliability theory to characterize distributions. Commun. Stat., Theory Methods 46 (2017), 8468-8478. DOI 10.1080/03610926.2016.1183784 | MR 3680770
[19] C. Kundu, A. K. Nanda, S. S. Maiti: Some distributional results through past entropy. J. Stat. Plann. Inference 140 (2010), 1280-1291. DOI 10.1016/j.jspi.2009.11.011 | MR 2581130 | Zbl 1186.60012
[20] A. K. Nanda: Characterization of distributions through failure rate and mean residual life functions. Stat. Probab. Lett. 80 (2010), 752-755. DOI 10.1016/j.spl.2010.01.006 | MR 2608812 | Zbl 1185.62031
[21] J. Navarro, Y. del Aguila, M. Asadi: Some new results on the cumulative residual entropy. J. Stat. Plann. Inference 140 (2010), 310-322. DOI 10.1016/j.jspi.2009.07.015 | MR 2568141 | Zbl 1177.62005
[22] L. Wang: Heat Capacity Bound, Energy Fluctuations and Convexity. Ph.D. Thesis, Yale University (2014). MR 3337578

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