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Generalized cumulative residual Tsallis entropy and its properties

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Abstract

In the literature, Shannon differential entropy has been widely used as a measure of uncertainty or dispersion. However, there exist alternative measures such as the cumulative residual Tsallis entropy that have shown promising results in various applications. Motivated by this, we introduce a new dispersion measure, the generalized cumulative residual Tsallis entropy, and thoroughly investigate its properties. Our approach is based on the Bayes risk of the mean residual lifetime function of the proportional hazards (PH) model under a suitable prior, leading to a novel and versatile measure of dispersion. We provide analytical representations, bounds, and stochastic ordering results for the new measure, highlighting its unique properties and potential applications. To further demonstrate the practicality of the mentioned approach, we propose a nonparametric estimator of the generalized CRTE based on the empirical distribution function and apply it to image processing. The findings offer new insights into statistical inference and data analysis, emphasizing the utility of the generalized cumulative residual Tsallis entropy as a powerful tool for addressing complex problems in various domains.

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References

  • Ardakani OM, Ebrahimi N, Soofi ES (2018) Ranking forecasts by stochastic error distance, information and reliability measures. Int Stat Rev 86(3):442–468

    Article  MathSciNet  MATH  Google Scholar 

  • Asadi M, Zohrevand Y (2007) On the dynamic cumulative residual entropy. J Stat Plan Inference 137(6):1931–1941

    Article  MathSciNet  MATH  Google Scholar 

  • Asadi M, Ebrahimi N, Soofi ES (2017) Connections of Gini, Fisher, and Shannon by Bayes risk under proportional hazards. J Appl Probab 54(4):1027

    Article  MathSciNet  MATH  Google Scholar 

  • Balakrishnan N, Leiva V, Sanhueza A, Cabrera E (2009) Mixture inverse gaussian distributions and its transformations, moments and applications. Statistics 43(1):91–104

    Article  MathSciNet  MATH  Google Scholar 

  • Collignon A, Maes F, Delaere D, Vandermeulen D, Suetens P, Marchal G (1995) Automated multi-modality image registration based on information theory. Inf Process Med Imaging 3:263–274

    Google Scholar 

  • Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B (Methodological) 34(2):187–202

    MathSciNet  MATH  Google Scholar 

  • Jones BL, Zitikis R (2003) Empirical estimation of risk measures and related quantities. N Am Actuarial J 7(4):44–54

    Article  MathSciNet  MATH  Google Scholar 

  • Kobbacy K, Fawzi B, Percy D, Ascher H (1997) A full history proportional hazards model for preventive maintenance scheduling. Qual Reliabil Eng Int 13(4):187–198

    Article  Google Scholar 

  • Makis V, Jardine A K (1992) Optimal replacement in the proportional hazards model. INFOR Inf Syst Oper Res 30(1):172–183

    MATH  Google Scholar 

  • Psarrakos G, Sordo MA (2019) On a family of risk measures based on proportional hazards models and tail probabilities. Insur Math Econ 86:232–240

    Article  MathSciNet  MATH  Google Scholar 

  • Rajesh G, Sunoj S (2019) Some properties of cumulative Tsallis entropy of order \(alpha\). Stat Pap 60(3):583–593

    Article  MATH  Google Scholar 

  • Rao M, Chen Y, Vemuri BC, Wang F (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50(6):1220–1228

    Article  MathSciNet  MATH  Google Scholar 

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

  • Tahmasebi S, Keshavarz A, Longobardi M, Mohammadi R (2020) A shift-dependent measure of extended cumulative entropy and its applications in blind image quality assessment. Symmetry 12(2):316

    Article  Google Scholar 

  • Toomaj A, Atabay HA (2022) Some new findings on the cumulative residual Tsallis entropy. J Comput Appl Math 400:113669

    Article  MathSciNet  MATH  Google Scholar 

  • Toomaj A, Di Crescenzo A (2020) Connections between weighted generalized cumulative residual entropy and variance. Mathematics 8(7):1072

    Article  Google Scholar 

  • Van den Poel D, Lariviere B (2004) Customer attrition analysis for financial services using proportional hazard models. Eur J Oper Res 157(1):196–217

  • Viola P, Wells WM III (1997) Alignment by maximization of mutual information. Int J Comput Vis 24(2):137–154

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, Gonbad Kavous, Iran

    Abdolsaeed Toomaj

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  1. Abdolsaeed Toomaj
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Correspondence to Abdolsaeed Toomaj.

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Communicated by Dan Goreac.

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Toomaj, A. Generalized cumulative residual Tsallis entropy and its properties. Comp. Appl. Math. 42, 330 (2023). https://doi.org/10.1007/s40314-023-02455-y

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  • DOI: https://doi.org/10.1007/s40314-023-02455-y

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