# Some Results on a Generalized Residual Entropy based on Order Statistics

## DOI:

https://doi.org/10.6092/issn.1973-2201/5500## Keywords:

generalized residual entropy, order statistics, parallel and series systems, maximum likelihood estimator## Abstract

In the present paper, we discuss some monotone properties of the GRE of order (α, β) in order statistics under various assumptions. It is shown that monotone properties are preserved under the formation of a parallel system but not under the formation of a series system. A counter example is presented. Bounds of the GRE of order statistics are obtained. The GRE of parallel and series systems are shown to be monotone function of the number of observations of a given sample. Numerical simulation is carried out for verification of the theoretical results. Maximum likelihood estimators of GRE of X, X_{1:n}and X

_{n:n}are obtained when independent data are drawn from exponential distribution.

## References

B. ABRAHAM, P. G. SANKARAN (2005). Renyi's entropy for residual lifetime distribution. Statistical Papers, 46, pp. 17–30.

M. ASADI, N. EBRAHIMI (2000). Residual entropy and its characterizations in terms of hazard function and mean residual life function. Statistics and Probability Letters, 49, pp. 263–269.

R. E. BARLOW, F. PROSCHAN (1981). Statistical Theory of Reliability and Life Testing: Probability Models. Rinehart and Winston, New York.

F. BELZUNCE, J. NAVARRO, J. M. RUIZ, Y. D. AGUILA (2004). Some results on residual entropy function. Metrika, 59, pp. 147–161.

T. M. COVER, J. A. THOMAS (2006). Elements of Information Theory. Wiley, New York.

H. A. DAVID, H. N. NAGARAJA (2003). Order Statistics. Wiley, New York.

D. DYER (1981). Structural probability bounds for the strong pareto laws. The Canadian Journal of Statistics, 9, pp. 71–77

N. EBRAHIMI (1996). How to measure uncertainty in the residual life distributions. Sankhya Series A, 58, pp. 48–57.

N. EBRAHIMI, S. N. U. A. KIRMANI (1996). Some results on ordering of survival functions through uncertainty. Statistics and Probability Letters, 11, pp. 167–

N. EBRAHIMI, F. PELLEREY (1995). New partial ordering of survival functions based on the notion of uncertainty. Journal of Applied Probability, 32, pp. 202–211.

J. N. KAPUR (1967). Generalized entropy of order alpha and type beta. Mathematical Seminar (Delhi), pp. 78–94.

J. N. KAPUR (1994). Measure of Information and Their Applications. Wiley Eastern Limited, New Delhi.

S. KAYAL (2015a). On generalized dynamic survival and failure entropies of order (_; _). Statistics and Probability Letters, 96, pp. 123–132.

S. KAYAL (2015b). Characterization based on generalized entropy of order statis- tics. Communications in Statistics Theory and Method, To appear.

S. KAYAL, P. VELLAISAMY (2011). Generalized entropy properties of records. Journal of Analysis, 19, pp. 25–40.

V. KUMAR, H. TANEJA (2011). Some characterization results on generalized cumulative residual entropy measure. Statistics and Probability Letters, 81, pp. 1072–1077.

X. LI, S. ZHANG (2011). Some new results on renyi entropy of residual life and inactivity time. Probability in the Engineering and Informational Sciences, 25, pp. 237–250.

M. MAHMOUDI, M. ASADI (2010). On the monotone behaviour of time dependent entropy of order alpha. Journal of Iranian Statistical Society, 9.

H. N. NAGARAJA (1990). Some reliability properties of order statistics. Communication in Statistics-Theory and Methods, 19, pp. 307–316.

S. NANDA, P. PAUL (2006). Some results on generalized residual entropy. Information Sciences, 176, pp. 27–47.

A. P. S. PHARWAHA, B. SINGH (2009). Shannon and non-shannon measures of entropy for statistical texture feature extraction in digitized mammograms. Proceedings of the World Congress on Engineering aand Computer Science, San Francisco, USA, 2, pp. 1286–1291.

A. RENYI (1961). On measures of entropy and information. Berkeley Symposium on Mathematics, Statistics and Probability, 1, pp. 547–461.

A. RENYI (2012). Probability Theory. Dover Publication.

M. SHAKED, J. G. SHANTHIKUMAR (2007). Stochastic Orders and Their Applications. Academic Press, New York.

C. SHANNON (1948). The mathematical theory of communication. Bell System Technical Journal, 27, pp. 379–423.

R. SMOLIKOVD, M. P. WACHOWIAK, G. D. TOURASSI, A. ELMAGHRABY, J. M. ZURADA (2002). Characterization of ultrasonic backscatter based on generalized entropy. Proceedings of the second joint Biomedical Engineering Society EMBS/BMES Conference, 2, pp. 953–954.

R. THAPLIYAL, H. C. TANEJA (2012). Generalized entropy of order statistics. Applied Mathematics, 3, pp. 1977–1982.

C. TSALLIS (1988). Possible generalization of boltzmann-gibbs statistics. Journal of Statistical Physics, 52, pp. 479–487.

R. S. VARMA (1966). Generalization of renyi's entropy of order a. Journal of Mathematical Sciences, 180, pp. 34–48.

S. ZAREZADEH, M. ASADI (2010). Results on residual renyi entropy of order statistics and record values. Information Sciences, 180, pp. 4195–4206.

## Downloads

## Published

## How to Cite

*Statistica*,

*74*(4), 383–402. https://doi.org/10.6092/issn.1973-2201/5500

## Issue

## Section

## License

Copyright (c) 2014 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.