Quantile-Based Generalized Entropy of Order (α, β) for Order Statistics
DOI:
https://doi.org/10.6092/issn.1973-2201/8525Keywords:
Quantile function, Quantile entropy, Reliability measures, Series and parallel systemAbstract
In the present paper, we propose a quantile version of generalized entropy measure for order statistics for residual and past lifetimes and study their properties. Lower and upper bound of the proposed measures are derived. It is shown that the quantile-based generalized information between i-th order statistics and parent random variable is distribution free. The uniform, exponential, generalized Pareto and finite range distributions, which are commonly used in the reliability modeling have been characterized in terms of the proposed entropy measure with extreme order statistics.
References
M. ABBASNEJAD, N. R. ARGHAMI (2011). Rényi entropy properties of order statistics. Communications in Statistics - Theory and Methods, 40, pp. 40-52.
B. C.ARNOLD,N. BALAKRISHNAN, H.N.NAGARAJA (1992). A First Course in Order Statistics. JohnWiley and Sons, New York.
M. A. K BAIG, J. G. DAR (2008). Generalized residual entropy function and its applications. European Journal of Pure and Applied Mathematics, 4, pp. 30-40.
S. BARATPOUR, A. KHAMMAR (2016). Tsallis entropy properties of order statistics and some stochastic comparisons. Journal of Statistical Research of Iran, 13, pp. 25-41.
S. BARATPOUR, A. KHAMMAR (2018). A quantile-based generalized dynamic cumulative measure of entropy. Communications in Statistics - Theory and Methods, 47, no. 13, pp. 3104-3117.
T. M. COVER, J. THOMAS (2006). Elements of Information Theory. John Wiley and Sons, Hoboken, New Jersey.
H. A. DAVID, H. N. NAGARAJA (2003). Order Statistics, Wiley, New York.
A. DI CRESCENZO, M. LONGOBARDI (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability, 39, pp. 434-440.
N. EBRAHIMI (1996). How to measure uncertainty in the residual lifetime distribution. Sankhya, Series A, 58, pp. 48-56.
N. EBRAHIMI, E. S. SOOFI, H. ZAHEDI (2004). Information properties of order statistics and spacings. IEEE Transactions on Information Theory, 50, no. 1, pp. 177-183.
W. GILCHRIST (2000). Statistical Modelling with Quantile Functions. Chapman and Hall/CRC, Boca Raton, FL.
S. KAYAL (2014). Some results on a generalized residual entropy based on order statistics. Statistica, 74, no. 4, 383-402.
S. KAYAL (2015). On generalized dynamic survival and failure entropies of order (α, β). Statistics and Probability Letters, 96, pp. 123-132.
S. KAYAL (2016). Characterization based on generalized entropy of order statistics. Communications in Statistics - Theory and Methods, 45, no. 15, pp. 4628-4636.
S. KAYAL, P. VELLAISAMY (2011). Generalized entropy properties of records. Journal of Analysis, 19, pp. 25-40.
S S. KULLBACK, R. A. LEIBLER (1951). On information and sufficiency. Annals of Mathematical Statistics, 22, pp. 79-86.
V. KUMAR (2015). Generalized entropy measure in record values and its applications. Statistics and Probability Letters, 106, pp. 46-51.
V. KUMAR, R. RANI (2018). Quantile approach of dynamic generalized entropy (divergence) measure. Statistica, 78, no. 2, pp. 105-126.
N. U. NAIR, P. G. SANKARAN, N. BALAKRISHNAN (2013). Quantile-Based Reliability Analysis. Springer, New York.
A. K. NANDA, P. PAUL (2006). Some properties of past entropy and their applications. Metrika, 64, pp. 47-61.
S. PARK (2005). Testing exponentiality based on Kullback-Leibler information with the type II censored data. IEEE Transactions Reliability, 54, pp. 22-26.
E. PARZEN (1979). Non parametric statistical data modelling. Journal of the American Statistical Association, 74, pp. 105-122.
A. RÉNYI (1961). On measures of entropy and information. Proceedings IV Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkley, pp. 547-561.
P. G. SANKARAN, S. M. SUNOJ, N. U. NAIR (2016). Kullback-Leibler divergence: A quantile approach. Statistics and Probability Letters, 111, pp. 72-79.
C. E. SHANNON (1948). A mathematical theory of communication. Bell System Technical Journal, 27, pp. 379-423.
S. M. SUNOJ, P. G. SANKARAN (2012). Quantile based entropy function. Statistics and Probability Letters, 82, pp. 1049-1053.
S. M. SUNOJ, A. S. KRISHNAN, P. G. SANKARAN (2017). Quantile based entropy of order statistics. Journal of the Indian Society for Probability and Statistics, 18, no. 1, pp. 1-17.
R. THAPLIYAL, H. C. TANEJA (2012). Generalized entropy of order statistics. Applied Mathematics, 3, pp. 1977-1982.
R. THAPLIYAL, H. C. TANEJA, V. KUMARM (2015). Characterization results based on non-additive entropy of order statistics. Physica A: Statistical Mechanics and its Applications, 417, pp. 297-303.
R. S. VARMA (1966). Generalization of Rényi's entropy of order α. Journal of Mathematical Sciences, 1, pp. 34-48.
K. M. WONG, S. CHEN (1990). The entropy of ordered sequences and order statistics. IEEE Transactions on Information Theory, 36, pp. 276-284.
S. ZAREZADEH, M. ASADI (2010). Results on residual Rényi entropy of order statistics and record values. Information Sciences, 180, no. 21, pp. 4195-4206.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Statistica
This journal is licensed under a Creative Commons Attribution 3.0 Unported License (full legal code).
Authors accept to transfer their copyrights to the journal.
See also our Open Access Policy.
