Abstract
In the present paper, we study the residual extropy using distribution function and quantile function approaches. We also investigate extropy in past lifetime in both approaches. Some characterizations and ageing properties of these extropy measures are proposed. Different stochastic orders based on the residual and past lifetime extropy are also presented.
Similar content being viewed by others
References
Alizadeh Noughabi, H., & Jarrahiferiz, J. (2018). On the estimation of extropy. Journal of Nonparametric Statistics,. https://doi.org/10.1080/10485252.2018.1533133.
Ayres, R. U., & Martinas, K. (1995). Waste potential entropy: The ultimate ecotoxic. Economie Appliquee, 48, 95–120.
Bain, Lee J. (1978). Statistical analysis of reliability and life-testing models. New York: Marcel-Dekker.
Bebbington, M., Lai, C. D., Murthy, D. N. P., & Zitikis, R. (2009). Modelling N-and W-shaped hazard rate functions without mixing distributions. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 223(1), 59–69.
Di Crescenzo, A., & Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability, 39(2), 434–440.
Ebrahimi-Sankhyā, N. (1996). How to measure uncertainty in the residual life time distribution. The Indian Journal of Statistics, Series A, 58(1), 48–56.
Good, I. J. (1979). Studies in the history of probability and statistics. XXXVII AM Turing’s statistical work in World War II. Biometrika, 66(2), 393–396.
Gore, A. P., Paranjape, S., Rajarshi, M. B., & Gadgil, M. (1986). Some methods for summarizing survivorship data in nonstandard situations. Biometrical journal, 28(5), 577–586.
Jose, J., & Abdul-Sathar, E. I. (2019). Residual extropy of k-record values. Statistics & Probability Letters,. https://doi.org/10.1016/j.spl.2018.10.019.
Lad, F., Sanfilippo, G., & Agro, G. (2015). Extropy: Complementary dual of entropy. Statistical Science, 30(1), 40–58.
Martinas, K., & Frankowicz, M. (2000). Extropy-reformulation of the entropy principle. Periodica Polytechnica Chemical Engineering, 44(1), 29–38.
Midhu, N. N., Sankaran, P. G., & Nair, N. U. (2013). A class of distributions with the linear mean residual quantile function and it’s generalizations. Statistical Methodology, 15, 1–24.
Midhu, N. N., Sankaran, P. G., & Nair, N. U. (2014). A class of distributions with linear hazard quantile function. Communications in Statistics-Theory and Methods, 43(17), 3674–3689.
Muliere, P., Parmigiani, G., & Polson, N. G. (1993). A note on the residual entropy function. Probability in the Engineering and Informational Sciences, 7(3), 413–420.
Nair, N. U., Sankaran, P. G., & Balakrishnan, N. (2013). Quantile-based Reliability Analysis. New York: Springer.
Nair, N. U., & Vineshkumar, B. (2011). Ageing concepts: An approach based on quantile function. Statistics & Probability Letters, 81(12), 2016–2025.
Nanda, A. K., Sankaran, P. G., & Sunoj, S. M. (2014). Renyi’s residual entropy: A quantile approach. Statistics & Probability Letters, 85, 114–121.
Qiu, G. (2017). The extropy of order statistics and record values. Statistics & Probability Letters, 120, 52–60.
Qiu, G. (2019). Further results on quantile entropy in the past lifetime. Probability in the Engineering and Informational Sciences, 33(1), 146–159.
Qiu, G., & Jia, K. (2018a). Extropy estimators with applications in testing uniformity. Journal of Nonparametric Statistics, 30(1), 182–196.
Qiu, G., & Jia, K. (2018b). The residual extropy of order statistics. Statistics & Probability Letters, 133, 15–22.
Shaked, M., & Shanthikumar, J. G. (2007). Stochastic orders. New York: Springer.
Sunoj, S. M., Krishnan, A. S., & Sankaran, P. G. (2017). Quantile-based entropy of order statistics. Journal of the Indian Society for Probability and Statistics, 18(1), 1–17.
Sunoj, S. M., & Sankaran, P. G. (2012). Quantile based entropy function. Statistics & Probability Letters, 82(6), 1049–1053.
Vineshkumar, B., Nair, N. U., & Sankaran, P. G. (2015). Stochastic orders using quantile-based reliability functions. Journal of the Korean Statistical Society, 44(2), 221–231.
Yang, J., Xia, W., & Hu, T. (2018). Bounds on eextropy with variational distance constraint. Probability in the Engineering and Informational Sciences,. https://doi.org/10.1017/S0269964818000098.
Yu, H.-L., & Wang, C.-H. (2013). Quantile-based bayesian maximum entropy approach for spatiotemporal modeling of ambient air quality levels. Environmental science & technology, 47(3), 1416–1424.
Acknowledgements
The first author is thankful to Kerala State Council for Science Technology and Environment (KSCSTE) for the financial support. The second author would like to thank the support of University Grants Commission, India, under the Special Assistance Programme.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Krishnan, A.S., Sunoj, S.M. & Unnikrishnan Nair, N. Some reliability properties of extropy for residual and past lifetime random variables. J. Korean Stat. Soc. 49, 457–474 (2020). https://doi.org/10.1007/s42952-019-00023-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42952-019-00023-x