Hostname: page-component-77c78cf97d-9lb97 Total loading time: 0 Render date: 2026-05-03T18:37:12.505Z Has data issue: false hasContentIssue false
  • English
  • Français

Varentropy of doubly truncated random variable

Part of: Communication, information Survival analysis and censored data Sufficiency and information

Published online by Cambridge University Press:  08 July 2022

Akash Sharma  and
Chanchal Kundu Open the ORCID record for Chanchal Kundu [Opens in a new window]
Akash Sharma
Affiliation:
Department of Mathematical Sciences, Rajiv Gandhi Institute of Petroleum Technology, Jais 229304, UP, India. E-mails: chanchal_kundu@yahoo.com, ckundu@rgipt.ac.in
Chanchal Kundu
Affiliation:
Department of Mathematical Sciences, Rajiv Gandhi Institute of Petroleum Technology, Jais 229304, UP, India. E-mails: chanchal_kundu@yahoo.com, ckundu@rgipt.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently, there is a growing interest to study the variability of uncertainty measure in information theory. For the sake of analyzing such interest, varentropy has been introduced and examined for one-sided truncated random variables. As the interval entropy measure is instrumental in summarizing various system and its components properties when it fails between two time points, exploring variability of such measure pronounces the extracted information. In this article, we introduce the concept of varentropy for doubly truncated random variable. A detailed study of theoretical results taking into account transformations, monotonicity and other conditions is proposed. A simulation study has been carried out to investigate the behavior of varentropy in shrinking interval for simulated and real-life data sets. Furthermore, applications related to the choice of most acceptable system and the first-passage times of an Ornstein–Uhlenbeck jump-diffusion process are illustrated.

Keywords

Doubly truncated random variableInterval varentropyParametric form of GFRVarentropy

MSC classification

Primary: 94A17: Measures of information, entropy
Secondary: 62B10: Information-theoretic topics 62N05: Reliability and life testing

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 3 , July 2023 , pp. 852 - 871
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

Arikan, E. (2016). Varentropy decreases under polar transform. IEEE Transactions on Information Theory 62: 3390–3400.CrossRefGoogle Scholar
Bobkov, S. & Madiman, M. (2011). Concentration of the information in data with log-concave distributions. The Annals of Probability 39: 1528–1543.CrossRefGoogle Scholar
Buono, F. & Longobardi, M. (2020). Varentropy of past lifetimes. Applied probability trust. Preprint arXiv:2008.07423.Google Scholar
Buono, F., Longobardi, M., & Szymkowiak, M. (2021). On generalized reversed aging intensity functions. Ricerche di Matematica. [Online First]. http://dx.doi.org/10.1007/s11587-021-00560-w.Google Scholar
Cacoullos, T. & Papathanasiou, V. (1989). Characterizations of distributions by variance bounds. Statistics and Probability Letters 7(5): 351–356.CrossRefGoogle Scholar
Dharmaraja, S., Di Crescenzo, A., Giorno, V., & Nobile, A.G. (2015). A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation. Journal of Statistical Physics 161: 326–345.CrossRefGoogle Scholar
Di Crescenzo, A. & Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability 39: 434–440.CrossRefGoogle Scholar
Di Crescenzo, A. & Paolillo, L. (2021). Analysis and applications of the residual varentropy of random lifetimes. Probability in the Engineering and Informational Sciences 35(3): 680–698.CrossRefGoogle Scholar
Di Crescenzo, A., Paolillo, L., & Suárez-Llorens, A. (2021). Stochastic comparisons, differential entropy and varentropy for distributions induced by probability density functions. Preprint arXiv:2103.11038v1.Google Scholar
Ebrahimi, N. & Pellerey, F. (1995). New partial ordering of survival functions based on the notion of uncertainty. Journal of Applied Probability 32(1): 202–211.CrossRefGoogle Scholar
Fradelizi, M., Madiman, M., & Wang, L. (2016). Optimal concentration of information content for log-concave densities. In High dimensional probability VII, vol. 71, pp. 45–60.CrossRefGoogle Scholar
Khorashadizadeh, M. (2019). A Quantile approach to the interval Shannon entropy. Journal of Statistical Research of Iran 15(2): 317–333.Google Scholar
Kundu, C. & Nanda, A.K. (2015). Characterizations based on measure of inaccuracy for truncated random variables. Statistical Papers 56(3): 619–637.CrossRefGoogle Scholar
Kundu, C. & Singh, S. (2019). On generalized interval entropy. Communications in Statistics – Theory and Methods 49(8): 1989–2007.CrossRefGoogle Scholar
Lee, E.T. & Wang, J.W. (2003). Statistical methods for survival data analysis. New York, USA: John Wiley and Sons.CrossRefGoogle Scholar
Maadani, S., Borzadaran, G.R.M., & Roknabadi, A.H.R. (2021). Varentropy of order statistics and some stochastic comparisons. Communications in Statistics – Theory and Methods. [Online First]. http://dx.doi.org/10.1080/03610926.2020.1861299.Google Scholar
Misagh, F. & Yari, G. (2011). On weighted interval entropy. Statistics and Probability Letters 81(2): 188–194.CrossRefGoogle Scholar
Misagh, F. & Yari, G. (2012). Interval entropy and informative distance. Entropy 14(3): 480–490.CrossRefGoogle Scholar
Moharana, R. & Kayal, S. (2020). Properties of Shannon entropy for double truncated random variables and its applications. Journal of Statistical Theory and Applications 19(2): 261–273.CrossRefGoogle Scholar
Navarro, J. & Ruiz, J.M. (1996). Failure rate functions for doubly truncated random variables. IEEE Transactions on Reliability 45(4): 685–690.CrossRefGoogle Scholar
Nourbakhsh, M. & Yari, G. (2014). Doubly truncated generalized entropy. In Proceedings of the 1st International Electronic Conference on Entropy and its Applications, 3–21 November.CrossRefGoogle Scholar
Raqab, M.Z., Bayoud, H.A., & Qiu, G. (2021). Varentropy of inactivity time of a random variable and its related applications. IMA Journal of Mathematical Control and Information. doi:10.1093/imamci/dnab033Google Scholar
Rioul, O. (2018). Renyi entropy power inequalities via normal transport and rotation. Entropy 20(9): 641.CrossRefGoogle ScholarPubMed
Shanker, R., Hagos, F., & Sujatha, S. (2015). On modeling of lifetimes data using exponential and Lindley distributions. Biometrics & Biostatistics International Journal 2(5): 1–9.CrossRefGoogle Scholar
Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal 27: 379–423, 623–656.CrossRefGoogle Scholar
Singh, S. & Kundu, C. (2018). On weighted Renyi's entropy for double-truncated distribution. Communications in Statistics – Theory and Methods 48(10): 2562–2579.CrossRefGoogle Scholar
Sunoj, S.M., Sankaran, P.G., & Maya, S.S. (2009). Characterizations of life distributions using conditional expectations of doubly (interval) truncated random variables. Communications in Statistics – Theory and Methods 38(9): 1441–1452.CrossRefGoogle Scholar