Abstract
The Shannon’s entropy function has a complementary dual function namely extropy and it facilitates the comparison of uncertainties of two random variables (see Lad et al. Stat Sci 30:40–58, 2015). Following the work of Lad et al. (Stat Sci 30:40–58, 2015), various generalizations/extensions of extropy measure are discussed in the literature analogous to that of Shannon’s entropy. Accordingly, a negative cumulative residual extropy is introduced by Tahmasebi and Toomaj (Commun Stat Theor Methods, 2020. https://doi.org/10.1080/03610926.2020.1831541). In the present work, we provide nonparametric kernel type estimators for the negative cumulative residual extropy based on the observations under study are dependent. Various properties including asymptotic properties of the proposed estimators are derived under suitable regularity conditions. A Monte-Carlo simulation study is carried out to find out the bias and mean squared error of the estimators.
Similar content being viewed by others
References
Al-Labadi, L., Berry, S.: Bayesian estimation of extropy and goodness of fit tests. J. Appl. Stat. (2020). https://doi.org/10.1080/02664763.2020.1812545
Becerra, A., de la Rosa, J.I., Gonzàlez, E., Pedroza, A.D., Escalante, N.I.: Training deep neural networks with non-uniform frame-level cost function for automatic speech recognition. Multimed. Tools Appl. 77, 27231–27267 (2018)
Cai, Z., Roussas., G.G.: Uniform strong estimation under \(\alpha \)-mixing, with rates. Stat. Probab. Lett. 15, 47–55 (1992)
Irshad, M.R., Maya, R.: Nonparametric estimation of past extropy under \(\alpha \)-mixing dependence. Ricerche mat. (2021). https://doi.org/10.1007/s11587-021-00570-8
Jahanshahi, S.M.A., Zarei, H., Khammar, A.H.: On cumulative residual extropy. Probab. Eng. Inf. Sci. 34, 605–625 (2020)
Lad, F., Sanfilippo, G., Agrò, G.: Extropy: complementary dual of entropy. Stat. Sci. 30, 40–58 (2015)
Masry, E.: Recursive probability density estimation for weakly dependent stationary process. IEEE Trans. Inf. Theory 32, 254–267 (1986)
Maya, R., Irshad, M.R.: Kernel estimation of residual extropy function under \(\alpha \)-mixing dependence condition. S. Afr. Stat. J. 53, 65–72 (2019)
Maya, R, Sathar, E.I.A., Rajesh, G., Nair, K.R.M.: Estimation of the Renyi’s residual entropy of order \(\alpha \) with dependent data. Stat. Pap. 55, 585–602 (2014)
Navarro, J., Psarrakos, G.: Characterizations based on generalized cumulative residual entropy functions. Commun. Stat.—Theory Methods 46, 1247–1260 (2017)
Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)
Qiu, G.: The extropy of order statistics and record values. Stat. Probab. Lett. 120, 52–60 (2017)
Qiu, G., Jia, K.: Extropy estimators with applications in testing uniformity. J. Nonparametric Stat. 30, 182–196 (2018)
Rao, M.: More on a new concept of entropy and information. J. Theor. Probab. 18, 967–981 (2005)
Rao, M., Chen, Y., Vemuri, B.C., Wang, F.: Cumulative residual entropy: a new measure of information. IEEE Trans. Inf. Theory 50, 1220–1228 (2004)
Raqab, M.Z., Qiu, G.: On extropy properties of ranked set sampling. Statistics 53, 210–226 (2019)
Rosenblatt, M.: Density estimates and Markov sequences. In: Puri, M.L. (ed.) Nonparametric Techniques in Statistical Inference. Oxford Cambridge University Press, London (1970)
Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42, 43–47 (1956)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)
Tahmasebi, S., Toomaj, A.: On negative cumulative extropy with applications. Commun. Stat.—Theory Methods (2020). https://doi.org/10.1080/03610926.2020.1831541
Wegman, E.J.: Nonparametric probability density estimation: I. A summary of available methods. Technometrics 14, 533–546 (1972)
Wolverton, C.T., Wagner, T.J.: Asymptotically optimal discriminant functions for pattern classification. IEEE Trans. Inf. Theory 15, 258–265 (1969)
Acknowledgements
The authors express their gratefulness for the constructive criticism of the learned referees which helped to improve considerably the revised version of the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Maya, R., Irshad, M.R. & Archana, K. Recursive and non-recursive kernel estimation of negative cumulative residual extropy under \(\alpha \)-mixing dependence condition. Ricerche mat 72, 119–139 (2023). https://doi.org/10.1007/s11587-021-00605-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-021-00605-0