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Bivariate extension of (dynamic) cumulative residual and past inaccuracy measures

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Abstract

In a recent paper, Kundu et al. (Metrika 79:335–356, 2016) study the notion of cumulative residual inaccuracy (CRI) and cumulative past inaccuracy (CPI) measures in univariate setup as a generalization of cumulative residual entropy and cumulative past entropy, respectively. Here we address the question of extending the definition of CRI (CPI) to bivariate setup and study their properties. We also prolong these measures to conditionally specified models of two components having possibly different ages or failed at different time instants called conditional CRI (CCRI) and conditional CPI (CCPI), respectively. We provide some bounds on using usual stochastic order and investigate several properties of CCRI (CCPI) including the effect of linear transformation. Moreover, we characterize some bivariate distributions.

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References

  • Ahmadi J, Di Crescenzo A, Longobardi M (2015) On dynamic mutual information for bivariate lifetimes. Adv Appl Probab 47(4):1157–1174

    Article  MathSciNet  Google Scholar 

  • Arnold BC (1995) Conditional survival models. In: Balakrishnan N (ed) Recent advances in life testing and reliability. CRC Press, Boca Raton, pp 589–601

    Google Scholar 

  • Arnold BC, Castillo E, Sarabia JM (1999) Conditional specification of statistical models. Springer, New York

    MATH  Google Scholar 

  • Asadi M, Zohrevand Y (2007) On the dynamic cumulative residual entropy. J Stat Plan Inference 137:1931–1941

    Article  MathSciNet  Google Scholar 

  • Barlow RE, Proschan F (1965) Mathematical theory of reliability. Wiley, New York

    MATH  Google Scholar 

  • Di Crescenzo A, Longobardi M (2009a) On cumulative entropies. J Stat Plan Inference 139:4072–4087

    Article  MathSciNet  Google Scholar 

  • Di Crescenzo A, Longobardi M (2009b) On cumulative entropies and lifetime estimations. In: International work-conference on the interplay between natural and artificial computation. Springer, Berlin, pp 132–141

  • Di Crescenzo A, Longobardi M (2013) Stochastic comparisons of cumulative entropies. In: Stochastic orders in reliability and risk. Lecture notes in statistics 208. Springer, New York, pp 167–182

  • Di Crescenzo A, Longobardi M (2015) Some properties and applications of cumulative Kullback–Leibler information. Appl Stoch Models Bus Ind 31(6):875–891

    Article  MathSciNet  Google Scholar 

  • Cahill ND, Schnabel JA, Noble JA, Hawkes DJ (2009) Overlap invariance of cumulative residual entropy measures for multimodal image alignment. In: SPIE medical imaging, 72590I-72590I. International society for optics and photonics

  • Ebrahimi N, Kirmani SNUA, Soofi ES (2007) Multivariate dynamic information. J Multivar Anal 98:328–349

    Article  MathSciNet  Google Scholar 

  • Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55:698–707

    Article  MathSciNet  Google Scholar 

  • Gumbel EJ (1961) Bivariate logistic distributions. J Am Stat Assoc 56:335–349

    Article  MathSciNet  Google Scholar 

  • Gupta RC, Gupta RD, Gupta PL (1998) Modeling failure time data by Lehman alternatives. Commun Stat Theory Methods 27:887–904

    Article  MathSciNet  Google Scholar 

  • Johnson NL, Kotz S (1975) A vector multivariate hazard rate. J Multivar Anal 5:53–66

    Article  MathSciNet  Google Scholar 

  • Kundu D, Gupta RD (2011) Absolute continuous bivariate generalized exponential distribution. AStA Adv Stat Anal 95:169–185

    Article  MathSciNet  Google Scholar 

  • Kundu A, Kundu C (2017) Bivariate extension of (dynamic) cumulative past entropy. Commun Stat Theory Methods 46(9):4163–4180

    Article  MathSciNet  Google Scholar 

  • Kundu C, Di Crescenzo A, Longobardi M (2016) On cumulative residual (past) inaccuracy for truncated random variables. Metrika 79(3):335–356

    Article  MathSciNet  Google Scholar 

  • Kumar V, Taneja HC (2015) Dynamic cumulative residual and past inaccuracy measures. J Stat Theory Appl 14(4):399–412

    MathSciNet  Google Scholar 

  • McGilchrist CA, Aisbett CW (1991) Regression with frailty in survival analysis. Biometrics 47:461–466

    Article  Google Scholar 

  • Mirhosseini SM, Amini M, Kundu D, Dolati A (2015) On a new absolutely continuous bivariate generalized exponential distribution. Stat Methods Appl 24:61–83

    Article  MathSciNet  Google Scholar 

  • Misagh F, Panahi Y, Yari GH, Shahi R (2011) Weighted cumulative entropy and its estimation. In: Proceedings of the IEEE international conference on quality and reliability (ICQR), pp 477–480

  • Nair NU, Asha G (2008) Some characterizations based on bivariate reversed mean residual life. ProbStat Forum 1:1–14

    MATH  Google Scholar 

  • Navarro J, del Aguila Y, Asadi M (2010) Some new results on the cumulative residual entropy. J Stat Plan Inference 140:310–322

    Article  MathSciNet  Google Scholar 

  • Navarro J, Sunoj SM, Linu MN (2011) Characterizations of bivariate models using dynamic Kullback–Leibler discrimination measures. Stat Probab Lett 81:1594–1598

    Article  MathSciNet  Google Scholar 

  • Navarro J, Sunoj SM, Linu MN (2014) Characterizations of bivariate models using some dynamic conditional information divergence measures. Commun Stat Theory Methods 43:1939–1948

    Article  MathSciNet  Google Scholar 

  • Nguyen HV, Müller E, Vreeken J, Keller F, Böhm K (2013) CMI: an information-theoretic contrast measure for enhancing subspace cluster and outlier detection. In: Proceedings of the SIAM international conference on data mining (SDM’13). Society for industrial and applied mathematics, Austin, Texas, USA, pp 198–206

  • Park S, Lim J (2015) On censored cumulative residual Kullback–Leibler information and goodness-of-fit test with type II censored data. Stat Pap 56(1):247–256

    Article  MathSciNet  Google Scholar 

  • Psarrakos G, Navarro J (2013) Generalized cumulative residual entropy and record values. Metrika 76(5):623–640

    Article  MathSciNet  Google Scholar 

  • Rajesh G, Abdul-Sathar EI, Nair KRM, Reshmi KV (2014) Bivariate extension of dynamic cumulative residual entropy. Stat Methodol 16:72–82

    Article  MathSciNet  Google Scholar 

  • Rao M, Chen Y, Vemuri BC, Wang F (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50:1220–1228

    Article  MathSciNet  Google Scholar 

  • Roy D (2002) A characterization of model approach for generating bivariate life distributions using reversed hazard rates. J Jpn Stat Soc 32(2):239–245

    Article  MathSciNet  Google Scholar 

  • Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(379–423):623–656

    Article  MathSciNet  Google Scholar 

  • Shi J, Cai Y, Zhu J, Zhong J, Wang F (2013) SEMG-based hand motion recognition using cumulative residual entropy and extreme learning machine. Med Biol Eng Comput 51(4):417–427

    Article  Google Scholar 

  • Sunoj SM, Linu MN (2012) Dynamic cumulative residual Renyi’s entropy. Statistics 41(1):41–56

    Article  MathSciNet  Google Scholar 

  • Sankaran PG, Nair NU (1993) A bivariate Pareto model and its applications to reliability. Naval Res Logist 40:1013–1020

    Article  MathSciNet  Google Scholar 

  • Sankaran PG, Sreeja VN (2007) Proportional hazards model for multivariate failure time data. Commun Stat Theory Methods 36:1627–1641

    Article  MathSciNet  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  Google Scholar 

  • Taneja HC, Kumar V (2012) On dynamic cumulative residual inaccuracy measure. In: Ao SI et al (ed) Proceedings of the world congress on engineering, vol 1, London, UK, pp 153–156

  • Wang FE, Vemuri BC, Rao M, Chen Y (2003) A new and robust information theoretic measure and its application to image alignment. In: Proceedings of the international conference on information processing in medical imaging, Ambleside, UK, pp 388–400

  • Zahedi H (1985) Some new classes of multivariate survival functions. J Stat Plan Inference 11:171–188

    Article  MathSciNet  Google Scholar 

  • Zardasht V, Parsi S, Mousazadeh M (2015) On empirical cumulative residual entropy and a goodness-of-fit test for exponentiality. Stat Pap 56(3):677–688

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors warmly thank the Editor-in-Chief of the journal and two anonymous reviewers for their constructive comments and suggestions leading to an improved version of the manuscript. The financial support (Ref. No. SR/FTP/MS-016/2012) rendered by the Department of Science and Technology, Government of India is acknowledged with thanks by C. Kundu for carrying out this research work.

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  1. Department of Mathematics, Rajiv Gandhi Institute of Petroleum Technology, Jais, UP, 229 304, India

    Amit Ghosh & Chanchal Kundu

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  1. Amit Ghosh
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  2. Chanchal Kundu
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Correspondence to Chanchal Kundu.

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Ghosh, A., Kundu, C. Bivariate extension of (dynamic) cumulative residual and past inaccuracy measures. Stat Papers 60, 2225–2252 (2019). https://doi.org/10.1007/s00362-017-0917-5

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