On Dynamic Generalized Measures of Inaccuracy
Keywords:Kerridge inaccuracy measure, Renyi entropy, Reliability measures, Characterization
Generalized information measures play an important role in the measurement of uncertainty of certain random variables, where the standard practice of applying ordinary uncertainty measures fails to fit. Based on the Renyi entropy and its divergence, we propose a generalized measure of inaccuracy of order α(≠1)>0
between two residual and past lifetime distributions of a system. We study some important properties and characterizations of these measures. A numerical example is given to illustrate the usefulness of the proposed measure.
B. ABRAHAM, P. G. SANKARAN (2006). Renyi’s entropy for residual lifetime distribution. Statistical Papers, 47, no. 1, pp. 17–29.
M. ASADI, N. EBRAHIMI, E. S. SOOFI (2005). Dynamic generalized information measures. Statistics and Probability Letters, 71, no. 1, pp. 85–98.
P. K. BHATIA (1999). On characterization of ‘useful’ inaccuracy of type . Soochow Journal of Mathematics., 25, no. 3, pp. 237–248.
P. K. BHATIA, H. C. TANEJA (1991). On characterization of quantitative-qualitative measure of inaccuracy. Information Sciences, 56, no. 1-3, pp. 143–149.
D. R. COX, D. OAKES (1984). Analysis of Survival Data. CRC Press, Boca Raton, Florida, USA.
A. DI CRESCENZO, M. LONGOBARDI (2004). A measure of discrimination between past lifetime distributions. Statistics and Probability Letters, 67, no. 2, pp. 173–182.
N. EBRAHIMI, S. N. U. A. KIRMANI (1996). A measure of discrimination between two residual life-time distributions and its applications. Annals of the Institute of Statistical Mathematics, 48, no. 2, pp. 257–265.
M. GIL (2011). On Rényi divergence measures for continuous alphabet sources. PhD Thesis. Queen’s University Kingston, Ontario, Canada.
J. N. KAPUR (1994). Measures of Information and their Applications. Wiley-Interscience, New York, USA.
J. N. KAPUR, H. K. KESAVAN (1992). Entropy optimization principles and their applications. In Entropy and Energy Dissipation in Water Resources, V.P. Singh and M. Fiorentino (eds.), Kluwer Academic Publishers, Springer, Netherlands, pp. 3–20.
D. F. KERRIDGE (1961). Inaccuracy and inference. Journal of the Royal Statistical Society. Series B, pp. 184–194.
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, R. SRIVASTAVA (2011). A dynamic measure of inaccuracy between two past lifetime distributions. Metrika, 74, no. 1, pp. 1–10.
C. KUNDU, A. K. NANDA (2015). Characterizations based on measure of inaccuracy for truncated random variables. Statistical Papers, 56, no. 3, pp. 619–637.
P. MULIERE, G. PARMIGIANI, N. G. POLSON (1993). A note on the residual entropy function. Probability in the Engineering and Informational Sciences, 7, no. 03, pp. 413–420.
P.NATH (1968). Entropy, inaccuracy and information. Metrika, 13, no. 1, pp. 136–148.
A. P. S. PHARWAHA, B. SINGH (2009). Shannon and non-Shannon measures of entropy for statistical texture feature extraction in digitizedmammograms. In Proceedings of the World Congress on Engineering and Computer Science. San Francisco, USA, vol. 2, pp. 20–22.
P. N. RATHIE, P. L. KANNAPPAN (1973). An inaccuracy function of type . Annals of the Institute of StatisticalMathematics, 25, no. 1, pp. 205–214.
A. RÉNYI (1961). On measures of entropy and information. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability. The Regents of the University of California, vol. 1, pp. 547–561.
C. E. SHANNON (1948). A note on the concept of entropy. The Bell System Technical Journal, 27, pp. 379–423.
B. D. SHARMA, R. AUTAR (1973). On characterization of a generalized inaccuracy measure in information theory. Journal of Applied Probability, 10, no. 02, pp. 464–468.
B. D. SHARMA, H. C. GUPTA (1976). Sub-additive measures of relative information and inaccuracy. Metrika, 23, no. 1, pp. 155–165.
R. SMOLIKOVA, M. WACHOWIAK, G. TOURASSI, A. ELMAGHRABY, J. ZURADA (2002). Characterization of ultrasonic backscatter based on generalized entropy. In Engineering in Medicine and Biology, 2002. 24th Annual Conference and the Annual Fall Meeting of the Biomedical Engineering Society EMBS/BMES Conference, 2002. Proceedings of the Second Joint. IEEE, Texas, USA, vol. 2, pp. 953–954.
H. C. TANEJA, V. KUMAR, R. SRIVASTAVA (2009). A dynamic measure of inaccuracy between two residual lifetime distributions. International Mathematical Forum, 4, no. 25, pp. 1213–1220.
H. C. TANEJA, R. K. TUTEJA (1986). Characterization of a quantitative-qualitative measure of inaccuracy. Kybernetika, 22, no. 5, pp. 393–402.
I. J. TANEJA, H. C. GUPTA (1978). On generalized measures of relative information and inaccuracy. Aplikace Matematiky, 23, no. 5, pp. 317–333.
C. TSALLIS (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52, no. 1, pp. 479–487.
R. S. VARMA (1966). Generalizations of Renyi’s entropy of order α. Journal of Mathematical Sciences, 1, no. 7, pp. 34–48.