Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T06:44:03.286Z Has data issue: false hasContentIssue false
  • English
  • Français

PROPERTIES FOR GENERALIZED CUMULATIVE PAST MEASURES OF INFORMATION

Part of: Survival analysis and censored data Communication, information Distribution theory - Probability

Published online by Cambridge University Press:  29 October 2018

Camilla Calì ,
Maria Longobardi  and
Jorge Navarro Open the ORCID record for Jorge Navarro [Opens in a new window]
Camilla Calì
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II Via Cintia, I-80126 Napoli, Italy E-mail: [email protected]
Maria Longobardi
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II Via Cintia, I-80126 Napoli, Italy E-mail: [email protected]
Jorge Navarro
Affiliation:
Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Shannon entropy based on the probability density function is a key information measure with applications in different areas. Some alternative information measures have been proposed in the literature. Two relevant ones are the cumulative residual entropy (based on the survival function) and the cumulative past entropy (based on the distribution function). Recently, some extensions of these measures have been proposed. Here, we obtain some properties for the generalized cumulative past entropy. In particular, we prove that it determines the underlying distribution. We also study this measure in coherent systems and a closely related generalized past cumulative Kerridge inaccuracy measure.

Keywords

Applied probabilityreliability theory

MSC classification

Primary: 94A17: Measures of information, entropy
Secondary: 62N05: Reliability and life testing 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 1 , January 2020 , pp. 92 - 111
Copyright
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.)

References

1.Asadi, M. & Zohrevand, Y. (2007). On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137: 1931–1941.Google Scholar
2.Barlow, R.E. & Proschan, F. (1965). Mathematical Theory of Reliability. New York: Wiley.Google Scholar
3.Burkschat, M. & Navarro, J. (2018). Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Probability in the Engineering and Informational Sciences 32: 246–274.Google Scholar
4.Di Crescenzo, A. & Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability 39: 434–440.Google Scholar
5.Di Crescenzo, A. & Longobardi, M. (2009). On cumulative entropies. Journal of Statistical Planning and Inference 139: 4072–4087.Google Scholar
6.Di Crescenzo, A. & Longobardi, M. (2009). On cumulative entropies and lifetime estimations. In Mira, J., Ferrandez, J.M., Alvarez Sanchez, J.R., Paz, F. & Toledo, J., (eds.), Methods and Models in Artificial and Natural Computation. IWINAC 2009, Lecture Notes in Computer Science Vol. 5601, Berlin: Springer-Verlag, pp. 132–141.Google Scholar
7.Di Crescenzo, A. & Longobardi, M. (2010). More on cumulative entropy. In Trappl, R., (eds.), Cybernetics and Systems 2010. Vienna: Austrian Society for Cybernetic Studies, pp. 181–186.Google Scholar
8.Di Crescenzo, A. & Longobardi, M. (2013). Stochastic comparisons of cumulative entropies. In Li, H. & Li, X., (eds.), Stochastic Orders in Reliability and Risk, In Honor of Professor Moshe Shaked. Lecture Notes in Statistics 208, New York: Springer, pp. 167–182.Google Scholar
9.Hairer, E., Norsett, P., & Wanner, G (2008). Solving Ordinary Differential Equations I. Berlin, Heildelberg: Springer-Verlag.Google Scholar
10.Kayal, S. (2016). On generalized cumulative entropies. Probability in the Engineering and Informational Sciences 30: 640–662.Google Scholar
11.Kerridge, D.F. (1961). Inaccuracy and Inference. Journal of the Royal Statistical Society, Series B 23: 184–194.Google Scholar
12.Li, X. & Lu, J. (2003). Stochastic comparisons on residual life and inactivity time of series and parallel systems. Probability in the Engineering and Informational Sciences 17: 267–275.Google Scholar
13.Longobardi, M. (2014). Cumulative measures of information and stochastic orders. Ricerche di Matematica 63: 209–223.Google Scholar
14.Nanda, A.K., Singh, H., Misra, N., & Paul, P. (2003). Reliability properties of reversed residual lifetime. Communication in Statistics Theory and Methods 32: 2031–2042. (with correction in Communication in Statistics Theory and Methods 33: 991–992, 2004).Google Scholar
15.Navarro, J., del Águila, Y. & Asadi, M. (2010). Some new results on the cumulative residual entropy. Journal of Statistical Planning and Inference 140: 310–322.Google Scholar
16.Navarro, J., del Águila, Y., Sordo, M.A., & Suárez-Llorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Applied Stochastic Models in Business and Industry 30: 444–454.Google Scholar
17.Navarro, J., del Águila, Y., Sordo, M.A., & Suárez-Llorens, A. (2016). Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Methodology and Computing in Applied Probability 18: 529–545.Google Scholar
18.Navarro, J. & Psarrakos, G. (2017). Characterization based on generalized cumulative residual entropy functions. Communication in Statistics Theory and Methods 46: 1247–1260.Google Scholar
19.Psarrakos, G. & Navarro, J. (2013). Generalized cumulative residual entropy and record values. Metrika 76: 623–640.Google Scholar
20.Rao, M. (2005). More on a new concept of entropy and information. Journal of Theoretical Probability 18: 967–981.Google Scholar
21.Rao, M., Chen, Y., Vemuri, B.C., & Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50: 1220–1228.Google Scholar
22.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.Google Scholar
23.Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal 27: 279–423.Google Scholar
24.Toomaj, S., Sunoj, S.M., & Navarro, J. (2017). Some properties of the cumulative residual entropy of coherent and mixed systems. Journal of Applied Probability 54: 379–393.Google Scholar
25.Wiener, N. (1961). Cybernetics (2nd Ed.). New York: The MIT Press and Wiley.Google Scholar