Hostname: page-component-594f858ff7-hf9kg Total loading time: 0 Render date: 2023-06-09T16:14:19.468Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "corePageComponentUseShareaholicInsteadOfAddThis": true, "coreDisableSocialShare": false, "useRatesEcommerce": true } hasContentIssue false


Published online by Cambridge University Press:  03 August 2018

Guoxin Qiu
Department of Business Administration Xinhua University of Anhui Hefei 230088, China and Department of Statistics and Finance, School of Management School of Business University of Science and Technology of China Hefei 230026, China E-mail:
Lechen Wang
Department of Statistics and Finance, School of Management School of Business University of Science and Technology of China Hefei 230026, China E-mail:
Xingyu Wang
Department of Statistics and Finance, School of Management School of Business University of Science and Technology of China Hefei 230026, China E-mail:


An expression of the extropy of a mixed system's lifetime was given firstly. Based on this expression, two mixed systems with same signature but with different components were compared. It was shown that the extropy of lifetime of a mixed system equals to that of its dual system if the lifetimes of the components have symmetric probability density function. Moreover, some bounds of the extropy of lifetimes of mixed systems were obtained and the concept of Jensen–extropy (JE) divergence of mixed systems was proposed. The JE divergence is non-negative and it can be used as an alternative information criteria for comparing mixed systems with homogeneous components. To illustrate the applications of JE divergence, some examples are addressed at the end of this paper.

Research Article
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


1.Agrò, G., Lad, F. & Sanfilippo, G. (2010). Sequentially forecasting economic indices using mixture linear combinations of EP distributions. Journal of Data Science 8: 101126.Google Scholar
2.Aliprantis, C.D. & Burkinshaw, O. (1981). Principles of real analysis. New York: Elsevier North-Holland Incorporated.Google Scholar
3.Asadi, M., Ebrahimi, N., Soofi, E.S. & Zohrevand, Y. (2016). Jensen–Shannon information of the coherent system lifetime. Reliability Engineering and System Safety 156: 244255.CrossRefGoogle Scholar
4.Bagai, L. & Kochar, S.C. (1986). On tail ordering and comparison of failure rates. Communications in Statistics–Theory and Methods 15: 13771388.CrossRefGoogle Scholar
5.Barlow, R. & Proschan, F. (1981). Statistical theory of reliability and life testing, probability model. New York: Holt, Rinehart and Winnston.Google Scholar
6.Boland, P.J. & Samaniego, F. (2004). The signature of a coherent system and its applications in reliability. In Soyer, R., Mazzuchi, T., & Singpurwalla, N.D., (eds.), Mathematical reliability: an expository perspective, Boston: Kluwer Publishers, pp. 129.Google Scholar
7.Capotorti, A., Regoli, G. & Vattari, F. (2010). Correction of incoherent conditional probability assessments. International Journal of Approximate Reasoning 51: 718727.CrossRefGoogle Scholar
8.Castet, J.F. & Saleh, J.H. (2010). Single versus mixture Weibull distributions for non-parametric satellite reliability. Reliability Engineering and System Safety 95: 295300.CrossRefGoogle Scholar
9.D'andrea, A. & De Sanctis, L., (2015). The Kruskal-Katona theorem and a characterization of system signatures. Journal of Applied Probability 52: 508518.CrossRefGoogle Scholar
10.Fashandi, F. & Ahmadi, J. (2012). Characterizations of symmetric distributions based on Renyi entropy. Statistics and Probability Letters 82: 798804.CrossRefGoogle Scholar
11.Gneiting, T. & Raftery, A.E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association 102: 359378.CrossRefGoogle Scholar
12.Jiang, R., Zuo, M.J. & Li, H.X. (1999). Weibull and inverse Weibull mixture models allowing negative weights. Reliability Engineering and System Safety 66: 227234.CrossRefGoogle Scholar
13.Lad, F., Sanfilippo, G. & Agrò, G. (2012). Completing the logarithmic scoring rule for assessing probability distributions. 11th Brazilian Meeting on Bayesian Statistics, Amparo, Brazil. In Stern, J.M., Lauretto, M.D., & Polpo, A. (eds.), AIP Conference Proceedings, Vol. 1490, pp. 1330.Google Scholar
14.Lad, F., Sanfilippo, G. & Agrò, G. (2015). Extropy: complementary dual of entropy. Statistical Science 30: 4058.CrossRefGoogle Scholar
15.Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. West Sussex: Wiley and Sons.Google Scholar
16.Murthy, D.N.P. & Jiang, R. (1997). Parametric study of sectional models involving two Weibull distributions. Reliability Engineering and System Safety 56: 151159.CrossRefGoogle Scholar
17.Qiu, G. (2017). The extropy of order statistics and record values. Statistics and Probability Letters 120: 5260.CrossRefGoogle Scholar
18.Qiu, G. & Jia, K. (2018). The residual extropy of order statistics. Statistics and Probability Letters 133: 1522.CrossRefGoogle Scholar
19.Qiu, G. & Jia, K. (2018). Extropy estimators with applications in testing uniformity. Journal of Nonparametric Statistics 30: 182196.CrossRefGoogle Scholar
20.Rao, M., Chen, Y., Vemuri, B.C. & Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50: 12201228.CrossRefGoogle Scholar
21.Samaniego, F.J. (2007). System signatures and their applications in engineering reliability. New York: Springer.CrossRefGoogle Scholar
22.Samaniego, F.J. & Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Advance in Applied Probability 48: 88111.CrossRefGoogle Scholar
23.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
24.Shaked, M. & Suarez-Llorens, A. (2003). On the comparison of reliability experiments based on the convolution order. Journal of the American Statistical Association 98: 693702.CrossRefGoogle Scholar
25.Shannon, C.E. (1948). A mathematical theory of communication. The Bell System Technical Journal 27: 379423.CrossRefGoogle Scholar
26.Toomaj, A. (2017). Rènyi entropy properties of mixed systems. Communications in Statistics–Theory and Methods 46: 906916.CrossRefGoogle Scholar
27.Toomaj, A. & Doostparast, M. (2014). A note on signature-based expressions for the entropy of mixed r–out–of–n systems. Naval Research Logistics 61: 202206.CrossRefGoogle Scholar
28.Toomaj, A. & Doostparast, M. (2016). On the Kullback Leibler information for mixed systems. International Journal of Systems Science 47: 24582465.CrossRefGoogle Scholar
29.Toomaj, A., Sunoj, S.M. & Navarro, J. (2017). Some properties of the cumulative residual entropy of coherent and mixed systems. Journal of Applied Probability 54: 379393.CrossRefGoogle Scholar