Abstract
In this paper, a family of mean past weighted (\(\hbox {MPW}_{\alpha }\)) distributions of order \(\alpha \) is introduced. For the construction of this family, the concepts of the mean inactivity time and cumulative \(\alpha \)-class past entropy are used. Distributional properties and stochastic comparisons with other known weighted distributions are given. Furthermore, an upper bound for the k-order moment of the random variables associated with the new family and a characterization result are obtained. Generalized discrete mixtures that involve \(\hbox {MPW}_{\alpha }\) distributions and other weighted distributions are also explored.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad, I.A., Kayid, M.: Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions. Probab. Eng. Inf. Sci. 19, 447–461 (2005)
Apostol, T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley, Boston (1974)
Asadi, M., Berred, A.: Properties and estimation of the mean past lifetime. Statistics 46, 405–417 (2012)
Bartoszewicz, J., Skolimowska, M.: Preservation of classes of life distributions and stochastic orders under weighting. Stat. Probab. Lett. 76, 587–596 (2006)
Block, H., Savits, T., Singh, H.: The reversed hazard rate function. Probab. Eng. Inf. Sci. 12, 69–90 (1998)
Calì, C., Longobardi, M., Ahmadi, J.: Some properties of cumulative Tsallis entropy. Phys. A 486, 1012–1021 (2017)
Di Crescenzo, A., Longobardi, M.: Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Probab. 39, 434–440 (2002)
Di Crescenzo, A., Longobardi, M.: On cumulative entropies. J. Stat. Plan. Inference 139, 4072–4087 (2009)
Di Crescenzo, A., Longobardi, M.: Stochastic comparisons of cumulative entropies. In: Li, H., Li, X. (eds.) Stochastic Orders in Reliability and Risk. Lecture Notes in Statistics, vol. 208, pp. 167–182. Springer, New York (2013)
Feizjavadian, S.H., Hashemi, R.: Mean residual weighted versus the length-biased Rayleigh distribution. J. Stat. Comput. Simul. 85, 2823–2838 (2015)
Havrda, J., Charvat, F.: Quantification method in classification processes: concept of structural \(\alpha \)-entropy. Kybernetika 3, 30–35 (1967)
Giorgi, G.M., Nadarajah, S.: Bonferroni and Gini indices for various parametric families of distributions. Metron 68, 23–46 (2010)
Gupta, R.C., Gupta, R.D.: Proportional reversed hazard rate model and its applications. J. Stat. Plan. Inference 137, 3525–3536 (2007)
Gupta, R.C., Kirmani, S.N.U.A.: The role of weighted distributions in stochastic modeling. Commun. Stat. Theory Methods 19, 3147–3162 (1990)
Jain, K., Singh, H., Bagai, I.: Relations for reliability measures of weighted distributions. Commun. Stat. Theory Methods 18, 4393–4412 (1989)
Kayid, M., Ahmad, I.A.: On the mean inactivity time ordering with reliability applications. Probab. Eng. Inf. Sci. 18, 395–409 (2004)
Longobardi, M.: Cumulative measures of information and stochastic orders. Ricerche Mat. 63, 209–223 (2014)
Misra, N., Gupta, N., Dhariyal, I.D.: Stochastic properties of residual life and inactivity time at a random time. Stoch. Models 24, 89–102 (2008)
Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)
Nanda, A.K., Jain, K.: Some weighted distributions results on univariate and bivariate cases. J. Stat. Plan. Inference 77, 169–180 (1999)
Nanda, A.K., Singh, H., Misra, N., Paul, P.: Reliability properties of reversed residual lifetime. Commun. Stat. Theory Methods 32, 2031–2042 (with correction in Commun. Stat. Theory Methods 33, 991–992 (2004)) (2003)
Navarro, J., del Aguila, Y., Ruiz, J.M.: Characterizations through reliability measures from weighted distributions. Stat. Pap. 42, 395–402 (2001)
Patil, G.P., Rao, C.R.: Weighted distributions and size-biased sampling with applications to wild-life populations and human families. Biometrics 34, 179–189 (1978)
Psarrakos, G., Economou, P.: On the generalized cumulative residual entropy weighted distributions. Commun. Stat. Theory Methods 46, 10914–10925 (2017)
Rajesh, G., Sunoj, S.M.: Some properties of cumulative Tsallis entropy of order \(\alpha \). Stat. Pap. 60, 933–943 (2019)
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)
Riabi, M.Y.A., Mohtashami Borzadaran, G.R., Yari, G.H.: \(\beta \)-entropy for Pareto-type distributions and related weighted distributions. Stat. Probab. Lett. 80, 1512–1519 (2010)
Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007)
Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)
Ullah, A.: Entropy, divergence and distance measures with econometric applications. J. Stat. Plan. Inference 49, 137–162 (1996)
Acknowledgements
C. Calì and M. Longobardi are partially supported by the GNAMPA research group of INDAM (Istituto Nazionale di Alta Matematica) and MIUR-PRIN 2017, Project “Stochastic Models for Complex Systems” (No. 2017JFFHSH).
Author information
Authors and Affiliations
Corresponding author
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
Calì, C., Longobardi, M. & Psarrakos, G. A family of weighted distributions based on the mean inactivity time and cumulative past entropies. Ricerche mat 70, 395–409 (2021). https://doi.org/10.1007/s11587-019-00475-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-019-00475-7