# Quantile Approach of Dynamic Generalized Entropy (Divergence) Measure

## DOI:

https://doi.org/10.6092/issn.1973-2201/7201## Keywords:

Quantile function, Varma's entropy, Divergence measure, Hazard quantile function, PH Model## Abstract

In the present paper, we propose a quantile version of generalized entropy measure for residual and past lifetimes and study their properties. Lower and upper bounds of the proposed measures are derived. Some of the quantile lifetime distributions have been characterized. We also introduce quantile versions of the generalized divergence measure of Varma between two residual and two past lifetime random variables. Some properties of this measure are studied and a characterization of the proportional (reversed) hazards model is given.

## References

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.

D. R. COX (1972). Regression models and life tables. Journal of the Royal Statistical Society, Series B, 34, pp. 187–220.

A. DI CRESCENZO, M. LONGOBARDI (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability, 39, pp. 434–440.

A. DI CRESCENZO, M. LONGOBARDI (2004). A measure of discrimination between past lifetime distributions. Statistics and Probability Letters, 67, pp. 173–182.

N. EBRAHIMI (1996). How to measure uncertainty in the residual lifetime distribution. Sankhya, Series A, 58, pp. 48–56.

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., pp. 383–402.

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

S. KAYAL (2015b). Some results on dynamic discrimination measures of order (alpha, beta). Hacettepe Journal of Mathematics and Statistics, 44, pp. 179–188.

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

S. KULLBACK, R. A. LEIBLER (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22, no. 1, pp. 79–86.

V. KUMAR, H. C. TANEJA (2011). Some characterization results on generalized cumulative residual entropy measure. Statistics and Probability Letters, 81, no. 8, pp. 72–77.

S. S. MAYA, S. M. SUNOJ (2008). Some dynamic generalized information measures in the context of weighted models. Statistica, 68, no.1, pp. 71–84.

N. U. NAIR, P. G. SANKARAN, N. BALAKRISHNAN (2013). Quantile-Based Reliability Analysis. Springer, New York.

N. U. NAIR, P. G. SANKARAN (2009). Quantile based reliability analysis. Communication in Statistics - Theory and Methods, 38, pp. 222–232.

A. K.NANDA, P. G. SANKARAN, S. M. SUNOJ (2014). Residual Renyi entropy: A Quantile approach. Statistics and Probability Letters, 85, pp. 114-121.

E. PARZEN (1979). Non parametric statistical data modelling. Journal of the American Statistical Association, 74, pp. 105–22.

A. RENYI (1961). On measure of entropy and information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1, pp. 547–561.

J. M. RUIZ, J. NAVARRO (1996). Characterizations based on conditional expectations of the doubled truncated distribution. Annals of the Institute of Statistical Mathematics, 48, no. 3, pp. 563–572.

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, M. N. LINU (2012). On bounds of some dynamic information divergence measures. Statistica, 72, no. 1, pp. 23–36.

S. M. SUNOJ, P. G. SANKARAN (2012). Quantile based entropy function. Statistics and Probability Letters, 82, pp. 1049–1053.

S. M. SUNOJ, P. G. SANKARAN, A. K. NANDA (2013). Quantile based entropy function in past lifetime. Statistics and Probability Letters, 83, pp. 366–372.

S. M. SUNOJ, P. G. SANKARAN, N. U. NAIR (2017). Quantile-based cumulative Kullaback-Leibler divergence. Statistics: A Journal of Theoretical and Applied Statistics, 52, no. 1, pp. 1–17.

R. S. VARMA (1966). Generalization of Renyi’s entropy of order $alpha$. Journal of Mathematical Sciences, 1, pp. 34–48.

## Downloads

## Published

## How to Cite

*Statistica*,

*78*(2), 105–126. https://doi.org/10.6092/issn.1973-2201/7201

## 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.