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Some distributional results through past entropy. (English) Zbl 1186.60012
Summary: Measure of uncertainty in past lifetime distribution plays an important role in the context of Information Theory, Forensic Science and other related fields. In this paper we provide characterizations of quite a few continuous and discrete distributions based on certain functional relationships among past entropy, reversed hazard rate and expected inactivity time. Based on past entropy, a conditional measure of uncertainty has been defined, which has helped in defining a new stochastic order and an ageing class. The properties of the stochastic order and those of the ageing class are also studied here.

##### MSC:
 60E15 Inequalities; stochastic orderings 62N05 Reliability and life testing 20B10 Characterization theorems for permutation groups
##### Keywords:
ageing property; inactivity time; stochastic order
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##### References:
 [1] Adams, G.; Watson, R., A discrete time parametric model for the analysis of failure time data, Australian journal of statistics, 31, 365-384, (1989) · Zbl 0707.62059 [2] Ahmad, I.A.; Kayid, M., Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions, Probability in the engineering and informational sciences, 19, 4, 447-461, (2005) · Zbl 1336.60027 [3] Ahmad, I.A.; Kayid, M.; Pellerey, F., Further results involving the MIT order and the IMIT class, Probability in the engineering and informational sciences, 19, 3, 377-395, (2005) · Zbl 1075.60010 [4] Azlarov, T.; Volodin, N., Characterization problems associated with exponential distribution, (1986), Springer Berlin · Zbl 0624.62020 [5] Badia, F.G., Berrade, M.D., 2008. On the reversed hazard rate and mean inactivity time of mixtures. In: Bedford, T., et al. (Eds.), Advances in Mathematical Modeling for Reliability, pp. 103-110. [6] Belzunce, F.; Navarro, J.; Ruiz, J.M.; del Aguila, Y., Some results on residual entropy function, Metrika, 59, 147-161, (2004) · Zbl 1079.62008 [7] Bracquemond, C.; Gaudoin, O., A survey on discrete lifetime distributions, International journal of reliability, quality and safety engineering, 10, 1, 69-98, (2003) [8] Chandra, N.K.; Roy, D., Some results on reversed hazard rate, Probability in the engineering and informational sciences, 15, 95-102, (2001) · Zbl 1087.62510 [9] Di Crescenzo, A.; Longobardi, M., Entropy-based measure of uncertainty in past lifetime distributions, Journal of applied probability, 39, 434-440, (2002) · Zbl 1003.62087 [10] Di Crescenzo, A.; Longobardi, M., A measure of discrimination between past lifetime distributions, Statistics and probability letters, 67, 173-182, (2004) · Zbl 1058.62088 [11] Galambos, J., Kotz, S., 1978. Characterization of Probability Distributions, a Unified Approach with an Emphasis on Exponential and Related Models. In: Lecture Notes on Mathematics, vol. 675, Springer, Berlin. · Zbl 0381.62011 [12] Goliforushani, S.; Asadi, M., On the discrete Mean past lifetime, Metrika, 68, 209-217, (2008) · Zbl 1433.62291 [13] Gupta, R.D.; Nanda, A.K., $$\alpha \text{- and} \beta \text{-entropies}$$ and relative entropies of distributions, Journal of statistical theory and applications, 1, 3, 177-190, (2002) [14] Hitha, N.; Nair, N.U., Characterization of some discrete models by properties of residual life function, Calcutta statistical association bulletin, 38, 219-223, (1989) · Zbl 0715.62023 [15] Kayid, M.; Ahmad, I.A., On the Mean inactivity time ordering with reliability applications, Probability in the engineering and informational sciences, 18, 3, 395-409, (2004) · Zbl 1059.62105 [16] Kemp, A.W., Classes of discrete lifetime distributions, Communications in statistics—theory & methods, 33, 3069-3093, (2004) · Zbl 1087.62016 [17] Li, X.; Xu, M., Some results about MIT order and IMIT class of life distributions, Probability in the engineering and informational sciences, 20, 3, 481-496, (2006) · Zbl 1122.60018 [18] Maiti, S.S.; Nanda, A.K., A loglikelihood-based shape measure of past lifetime distribution, Calcutta statistical association bulletin, 61, 303-320, (2009) [19] Nair, N.U.; Hitha, N., Characterization of discrete models by distribution based on their partial sums, Statistics and probability letters, 8, 335-337, (1989) · Zbl 0676.62012 [20] Nair, K.R.M.; Rajesh, G., Characterization of probability distributions using the residual entropy function, Journal of Indian statistical association, 36, 157-166, (1998) [21] Nanda, A.K.; Das, S., Study on R-norm residual entropy, Calcutta statistical association bulletin, 58, 231-232, 1-12, (2006) [22] Nanda, A.K.; Paul, P., Some properties of past entropy and their applications, Metrika, 64, 47-61, (2006) · Zbl 1104.94007 [23] Nanda, A.K.; Paul, P., Some results on generalized past entropy, Journal of statistical planning and inference, 136, 3659-3674, (2006) · Zbl 1098.94014 [24] Nanda, A.K.; Paul, P., Some results on generalized residual entropy, Information sciences, 176, 27-47, (2006) · Zbl 1093.94016 [25] Nanda, A.K.; Sengupta, D., Discrete life distributions with decreasing reversed hazard, Sankhyā A, 67, 1, 106-125, (2005) · Zbl 1192.90055 [26] Navarro, J.; del Aguila, Y.; Asadi, M., Some new results on the cumulative residual entropy, Journal of statistical planning and inference, 140, 1, 310-322, (2010) · Zbl 1177.62005 [27] Navarro, J.; Franco, M.; Ruiz, J.M., Characterization through moments of the residual life and conditional spacings, Sankhyā, 60, 36-38, (1998) · Zbl 0977.62010 [28] Roy, D.; Gupta, R.P., Classification of discrete lives, Microelectronics reliability, 32, 10, 1459-1473, (1992) [29] Ruiz, J.M.; Navarro, J., Characterization of discrete distributions using expected values, Statistical papers, 36, 237-252, (1995) · Zbl 0833.62011 [30] Ruiz, J.M.; Navarro, J., Characterizations based on conditional expectations of the double truncated distribution, Annals of the institute of statistical mathematics, 48, 563-572, (1996) · Zbl 0925.62059 [31] Sankaran, P.G.; Gupta, R.P., Characterization of lifetime distributions using measure of uncertainty, Calcutta statistical association bulletin, 49, 195-196, 159-166, (1999) · Zbl 1110.62307 [32] Sengupta, D.; Chatterjee, A.; Chakraborty, B., Reliability bounds and other inequalities for discrete life distributions, Microelectronics reliability, 35, 12, 1473-1478, (1995) [33] Shaked, M.; Shanthikumar, J.G.; Valdez-Torres, J.B., Discrete probabilistic orderings in reliability theory, Statistica sinica, 4, 567-579, (1994) · Zbl 0823.62082 [34] Shaked, M.; Shanthikumar, J.G.; Valdez-Torres, J.B., Discrete hazard rate functions, Computers and operations research, 22, 391-402, (1995) · Zbl 0822.90073 [35] Shannon, C.E., A mathematical theory of communications, Bell system technical journal, 27, 379-423, (1948), 623-656 · Zbl 1154.94303 [36] Xekalaki, E., Hazard functions and life distributions in discrete time, Communications in statistics—theory & methods, 12, 2503-2509, (1983) · Zbl 0552.62008 [37] Xekalaki, E.; Dimaki, C., Identifying the Pareto and Yule distributions by properties of their reliability measures, Journal of statistical planning and inference, 131, 231-252, (2005) · Zbl 1061.62020
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