Abstract
The representation of the entropy in terms of the hazard function and its extensions have been studied by many authors including Teitler et al. (IEEE Trans Reliab 35:391–395, 1986). In this paper, we consider a representation of the Kullback–Leibler information of the first \(r\) order statistics in terms of the relative risk (Park and Shin in Statistics, 2012), the ratio of hazard functions, and extend it to the progressively Type II censored data. Then we study the change in Kullback–Leibler information of the first \(r\) order statistics according to \(r\) and discuss its relation with Fisher information in order statistics.
Similar content being viewed by others
References
Abo-Eleneen ZA (2011) The entropy of progressively censored samples. Entropy 13:437–449
Abbasnejad M, Arghami NR (2011) Renyi entropy properties of order statistics. Commun Stat Theory Methods 40:40–52
Balakrishnan N, Aggarawala R (2000) Progressive censoring: theory, methods, and applications. Birkhauser, Boston, MA
Balakrishnan N, Rad AH, Arghami NR (2007) Testing exponentiality based on Kullback–Leibler information with progressively Type-II censored data. IEEE Trans Relib 56:301–307
Cole RH (1951) Relations between moments of order statistics. Ann Math Stat 22:308–310
David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, New York
Ebrahimi N, Soofi ES, Zahedi H (2004) Information properties of order statistics and spacings. IEEE Trans Inf Theory 50:177–183
Efron B, Johnstone I (1990) Fisher information in terms of the hazard rate. Ann Stat 18:38–62
Kullback S (1959) Information theory and statistics. Wiley, NY
Park S (1995) The entropy of consecutive order statistics. IEEE Trans Inf Theory 41:2003–2007
Park S (1996) Fisher information in order statistics. J Am Stat Assoc 91:385–390
Park S (2003) On the asymptotic Fisher information in order statistics. Metrika 57:71–80
Park S (2005) Testing exponentiality based on Kullback–Leiber information with the Type II cnesored data. IEEE Trans Reliab 54:22–26
Park S, Shin M (2013) Kullback–Leibler information of Type I censored variable and its application. Statistics (to appear)
Rad AH, Yousefzadeh F, Balakrishnan N (2011) Goodness-of-fit test based on Kullback–Leibler information for progressively Type-II censored data. IEEE Trans Reliab 60:570–579
Teitler S, Rajagopal AK, Ngai KL (1986) Maximum entropy and reliability distributions. IEEE Trans Reliab 35:391–395
Wong KM, Chen S (1990) The entropy of ordered sequences and order statistics. IEEE Trans Inf Theory 36:276–284
Zahedi H, Shakil M (2006) Properties of entropies of record values in reliability and life testing context. Commun Stat Theory Methods 35:997–1010
Zheng G, Park S (2004) On the Fisher information in multiply censored and progressively censored data. Commun Stat Theory Methods 33:1821–1835
Acknowledgments
The author is grateful to anonymous referees and an associate editor for making some useful comments on an earlier version of this manuscript. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2011-0029104) and the Korea government (MEST) (No. 2012-004905).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Park, S. On Kullback–Leibler information of order statistics in terms of the relative risk. Metrika 77, 609–616 (2014). https://doi.org/10.1007/s00184-013-0455-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-013-0455-7