Review Articles

Some results on quantile-based Shannon doubly truncated entropy

Vikas Kumar ,

Department of Applied Sciences, UIET, M. D. University, Rohtak, India

Gulshan Taneja ,

Department of Mathematics, M. D. University, Rohtak, India

Samsher Chhoker

Department of Mathematics, M. D. University, Rohtak, India

Pages 59-70 | Received 11 Apr. 2018, Accepted 20 Feb. 2019, Published online: 15 Mar. 2019,
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Sunoj et al. [(2009). Characterization of life distributions using conditional expectations of doubly (Intervel)truncated random variables. Communications in Statistics – Theory and Methods38(9), 1441–1452] introduced the concept of Shannon doubly truncated entropy in the literature. Quantile functions are equivalent alternatives to distribution functions in modelling and analysis of statistical data. In this paper, we introduce quantile version of Shannon interval entropy for doubly truncated random variable and investigate it for various types of univariate distribution functions. We have characterised certain specific lifetime distributions using the measure proposed. Also we discuss one fascinating practical example based on the quantile data analysis.


  1. Baratpour, S., & Khammar, A. H. (2018). A quantile-based generalized dynamic cumulative measure of entropy. Communications in Statistics – Theory and Methods47(13), 3104–3117. doi: 10.1080/03610926.2017.1348520 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  2. Di Crescenzo, A., & Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability39, 434–440. doi: 10.1239/jap/1025131441 [Crossref][Web of Science ®], [Google Scholar]
  3. Di Crescenzo, A., & Longobardi, M. (2004). A measure of discrimination between past lifetime distributions. Statistics & Probability Letters67, 173–182. doi: 10.1016/j.spl.2003.11.019 [Crossref][Web of Science ®], [Google Scholar]
  4. Ebrahimi, N. (1996). How to measure uncertainty in the residual life distributions. Sankhya Series A58, 48–57. [Google Scholar]
  5. Gilchrist, W. (2000). Statistical modelling with quantile functions. Boca Raton, FL: Chapman and Hall/CRC. [Crossref], [Google Scholar]
  6. Gong, W., Yang, D., Gupta, H. V., & Nearing, G. (2014). Estimating information entropy for hydrological data: One-dimensional case. Water Resources Research50, 5003–5018. doi: 10.1002/ 2014WR015874 [Crossref][Web of Science ®], [Google Scholar]
  7. Hankin, R. K. S., & Lee, A. (2006). A new family of non-negative distributions. Australian and New Zealand Journal of Statistics48, 67–78. doi: 10.1111/j.1467-842X.2006.00426.x [Crossref][Web of Science ®], [Google Scholar]
  8. Kayal, S., & Moharana, R. (2016). Some Results on a doubly truncated generalized discrimination measure. Applications of Mathematics61, 585–605. doi: 10.1007/s10492-016-0148-4 [Crossref][Web of Science ®], [Google Scholar]
  9. Khorashadizadeh, M., Rezaei Roknabadi, A. H., & Mohtashami Borzadaran, G. R. (2013). Mohtashami Borzadaran Doubly truncated (interval) cumulative residual andpast entropy. Statistics & Probability Letters83, 1464–1471. doi: 10.1016/j.spl.2013.01.033 [Crossref][Web of Science ®], [Google Scholar]
  10. Kumar, V. R. (2018). A quantile approach of Tsallis entropy for order statistics. Physica A: Statistical Mechanics and its Applications503, 916–928. doi: 10.1016/j.physa.2018.03.025 [Crossref][Web of Science ®], [Google Scholar]
  11. Kundu, C. (2017). On weighted measure of inaccuracy for doubly truncated random variables. Communications in Statistics – Theory and Methods46, 3135–3147. doi: 10.1080/03610926.2015.1056365 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  12. Misagh, F., & Yari, G. (2011). On weighted interval entropy. Statistics & Probability Letters29, 167–176. [Google Scholar]
  13. Misagh, F., & Yari, G. H. (2012). Interval entropy and Informative Distance. Entropy14, 480–490. doi: 10.3390/e14030480 [Crossref][Web of Science ®], [Google Scholar]
  14. Nair, K. R. M., & Rajesh, G. (2000). Geometric vitality function and its applications to reliability. IAPQR Transactions25, 1–8. [Google Scholar]
  15. Nair, N. U., Sankaran, P. G., & Balkrishanan, N. (2013). Quantile based reliability analysis. Statistics for industry and technology. New York, NY: Springer Science+Business Media. [Google Scholar]
  16. Nair, N. U., Sankaran, P. G., & Vinesh Kumar, B. (2012). Modeling lifetimes by quantile functions using Parzen's score function. Statistics-A Journal of Theoretical and Applied Statistics46(6), 799–811. [Google Scholar]
  17. Nanda, A. K., Sankaran, P. G., & Sunoj, S. M. (2014). Renyi's residual entropy: A quantile approach. Statistics & Probability Letters85, 114–121. doi: 10.1016/j.spl.2013.11.016 [Crossref][Web of Science ®], [Google Scholar]
  18. Qiu, G. (2018). Further results on the residual quantile entropy. Communications in Statistics – Theory and Methods47(13), 3092–3103. doi: 10.1080/03610926.2017.1348519 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  19. Ruiz, J. M., & Navarro, J. (1996). Characterizations based on conditional expectations of the doubled truncated distribution. Annals of the Institute of Statistical Mathematics48(3), 563–572. doi: 10.1007/BF00050855 [Crossref][Web of Science ®], [Google Scholar]
  20. Sankaran, P. G., & Gupta, R. P. (1999). Characterization of lifetime distributions using measure of uncertainty. Calcutta Statistical Association Bulletin49, 195–196. doi: 10.1177/0008068319990303 [Crossref], [Google Scholar]
  21. Sankaran, P. G., & Sunoj, S. M. (2004). Identification of models using failure rate and mean residual life of doubly truncated random variables. Statistical Papers45, 97–109. doi: 10.1007/BF02778272 [Crossref][Web of Science ®], [Google Scholar]
  22. Sankaran, P. G., & Sunoj, S. M. (2017). Quantile based cumulative entropies. Communications in Statistics – Theory and Methods46(2), 805–814. doi: 10.1080/03610926.2015.1006779 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  23. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal27, 379–423. doi: 10.1002/j.1538-7305.1948.tb01338.x [Crossref][Web of Science ®], [Google Scholar]
  24. Simpson, J., Olsen, A., & Eden, J. (1975). A Baysian analysis of a multiplicative treatment effect in weather modification. Technometrics17, 161–166. doi: 10.2307/1268346 [Crossref][Web of Science ®], [Google Scholar]
  25. Sunoj, S. M., & Sankaran, P. G. (2012). Quantile based entropy function. Statistics & Probability Letters82, 1049–1053. doi: 10.1016/j.spl.2012.02.005 [Crossref][Web of Science ®], [Google Scholar]
  26. Sunoj, S. M., Sankaran, P. G., & Maya, S. S. (2009). Chararacterization of life distributions using conditional expectations of doubly (Intervel)truncated random variables. Communications in Statistics – Theory and Methods38(9), 1441–1452. doi: 10.1080/03610920802455001 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  27. Sunoj, S. M., Sankaran, P. G., & Nanda, A. K. (2013). Quantile based entropy function in past lifetime. Statistics & Probability Letters83, 366–372. doi: 10.1016/j.spl.2012.09.016 [Crossref][Web of Science ®], [Google Scholar]
  28. van Staden, P. J., & Loots, M. T. (2009). L-moment estimation for the generalized lambda distribution. Third annual ASEARC conference, New Castle. [Google Scholar]