Characterizations of some Discrete Distributions and Upper Bounds on Discrete Residual Varentropy

Document Type : Original Article

Author
Faranak Goodarzi
Faculty of Mathematics, Department of Statistics, University of Kashan
10.22034/jirss.2022.706994
Abstract
In this article, we obtain an upper bound for the variance of a function of the residual life random variable for discrete lifetime distributions. As a special case, we find an upper bound for residual varentropy. Moreover we characterize some discrete distributions by Cauchy-Schwarz inequality. We also get new expressions, bounds and stochastic comparisons involving measures in reliability and information theory.
Keywords
Characterization
Discrete aging Intensity
Discrete Cumulative Residual Entropy
Discrete Mean Residual Life Function
Discrete Residual Varentropy
Discrete Variance Residual Life Function
Hazard Rate
Almalki, S.J., Nadarajah, S. (2014). Modifications of the Weibull distribution: A review, Reliability Engineering and System Safety, 124, 32–55.
Baratpour, S. and Bami, Z. (2012). On the Discrete Cumulative Residual Entropy. Journal of the iranian statistical society, 11, 203-215.
Cacoullos, T. (1982). On upper and lower bounds for the variance of the function of a random variable. Annals of Probability, 10, 799-809.
Cacoullos, T. and Papathanasiou, V. (1985). On upper bounds for the variance of functions of random variables. Statistics and Probability Letters, 3, 175-184.
Cover, T. M. and Thomas J. A. (2006). elements of information theory. Second Edition. Wiley Interscience.
Di Crescenzo, A. and Paolillo, L. (2021). Analysis and applications of the residual varentropy of random lifetimes. Probability in the Engineering and Informational Sciences, 35, 680-698.
Gupta, P. L. (2015). Properties of reliability functions of discrete distributions. Communicatons in Statistics-Theorey and Methods, 44(19), 4114-4131.
Gupta, R. C. (2006). Variance residual life function in reliability studies. Metron International Journal of Statistics LXIV, 3, 343-355.
Gupta, R. C. and Kirmani, S.N.U.A. (1998). Residual Life Function in Reliability Studies. In: Fronteriors of Reliability, Basu A.P., Basu S.K. and Mukhopadhyay S. (eds.), World Scientific, New Jersey.
Gupta, R. C., Kirmani, S.N.U.A. and Launer, R. L. (1987). On life distributions having monotone residual variance. Probability in the Engineering and Informational Sciences, 1, 299-307.
Gupta, P. L., Gupta, R. C. and Tripathi, R. C. (1997). On the monotonic properties of discrete failure rates. Journal of Statistical Planning and Inference, 65(2), 255-268.
Kapodistria, S., and Psarrakos, G. (2012). Probability in the Engineering and Informational Sciences, 26(1), 129-146.
Kemp, A.W. (2004). Classes of discrete lifetime distributions. Communication in Statistic Theory and Methods, 33, 3069-3093.
Rajesh, G. and Sunoj, S. M. (2019). Some properties of cumulative Tsallis entropy of order {alpha}. Statistical Papers, 60(3), 933–943.
Szymkowiak, M. and Iwi ´nska, M. (2016). Characterizations of Discrete Weibull related distributions. Statistics and Probability Letters, 111, 41-48.
Toomaj, A., and Di Crescenzo, A. (2020). Generalized entropies, variance and applications. Entropy, 22, 709.
Xie, M., Gaudoin, O., Bracqemond, C. (2002). Redefining failure rate function for discrete distribution. International Journal of Reliability, Quality Saftey Enginearing, 9(3), 275–285.
Journal of the Iranian Statistical Society
Volume 21, Issue 2
December 2022
Pages 233-250

  • Receive Date 26 November 2021
  • Accept Date 15 March 2023