Statistical Inference on the CR Extropy and DCR Extropy of Equilibrium distribution of order r

Document Type : Original Article

Authors
Rajesh Ganapathi
Sajily V S
Department of Statistics, Cochin University of Science and Technology, Cochin 682 022, Kerala, India.
10.22034/jirss.2024.2005360.1025
Abstract
This study develops estimators for Cumulative Residual Extropy (CR Extropy) and Dynamic Cumulative Residual Extropy (DCR Extropy) of the equilibrium distribution of order $r$ of Lomax distribution under progressively Type-II censored data. Maximum likelihood and Bayesian estimators are obtained. The Bayesian estimators are evaluated based on LINEX loss functions. Non-informative and informative priors are considered for unknown parameters. By using Lindley's approximation and the Importance Sampling Methods, it is possible to approximate the closed-form expressions of the Bayesian estimators. The confidence intervals for the estimators are calculated using the normal approximation method and bootstrap algorithms. We have conducted simulation studies to evaluate how well the proposed estimators perform. Additionally, we have tested the practical utility of these estimators using a real data.
Keywords
Bayesian Estimation
CR Extropy
DCR Extropy
Lomax Distribution
MLE
Progressively Type-II Censoring
Subjects
Abd-Elfattah, A.M., Alaboud, F.M. and Alharby, A.H. (2007). On sample size estimation for Lomax distribution,  Australian Journal of Basic and Applied Sciences,  1(4), 373--378.
Asgharzadeh, A. and Valiollahi, R. (2010). Estimation of the scale parameter of the Lomax distribution under progressive censoring,  International Journal of Statistics & Economics,  6(S11), 37--48.
Cramer, E. and Schmiedt, A.B. (2011). Progressively Type-II censored competing risks data from Lomax distributions,  Computational Statistics & Data Analysis,  55(3), 1285--1303.
Helu, A. and Samawi, H. (2021). Statistical analysis based on adaptive progressive hybrid censored data from Lomax distribution,  Statistics, Optimization & Information Computing,  9(4), 789.
Lomax, K.S. (1954). Business failures: Another example of the analysis of failure data,  Journal of the American Statistical Association,  49(268), 847--852.
Bryson, M.C. (1974). Heavy-tailed distributions: properties and tests,  Technometrics,  16(1), 61--68.
Chahkandi, M. and Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate,  Computational Statistics & Data Analysis,  53(12), 4433--4440.
Johnson, N.L., Kotz, S. and Balakrishan, N. (1994). Continuous Univariate Distributions, Volume 2. Wiley.
Rajab, M., Aleem, M., Nawaz, T. and Daniyal, M. (2013). On five parameter beta Lomax distribution,  Journal of Statistics,  20(1).
El-Bassiouny, A.H., Abdo, N.F. and Shahen, H.S. (2015). Exponential Lomax distribution,  International Journal of Computer Applications,  121(13).
Cordeiro, G.M., Ortega, E.M.M. and Popović, B.V. (2015). The gamma-Lomax distribution,  Journal of Statistical Computation and Simulation,  85(2), 305--319.
Al-Zahrani, B. and Sagor, H. (2014). The Poisson-Lomax distribution,  Revista Colombiana de Estadística,  37(1), 225--245.
Tahir, M.H., Cordeiro, G.M., Mansoor, M. and ZUBAİR, M. (2015). The Weibull-Lomax distribution: properties and applications,  Hacettepe Journal of Mathematics and Statistics,  44(2), 455--474.
Kilany, N.M. (2016). Weighted Lomax distribution,  SpringerPlus,  5(1), 1--18.
Balakrishnan, N., Balakrishnan, N. and Aggarwala, R. (2000). Progressive censoring: theory, methods, and applications. Springer Science & Business Media.
Kundu, D. and Joarder, A. (2006). Analysis of Type-II progressively hybrid censored data,  Computational Statistics & Data Analysis,  50(10), 2509--2528.
Helu, A., Samawi, H. and Raqab, M.Z. (2015). Estimation on Lomax progressive censoring using the EM algorithm,  Journal of Statistical Computation and Simulation,  85(5), 1035--1052.
Harris, C.M. (1968). The Pareto distribution as a queue service discipline,  Operations Research,  16(2), 307--313.
Ebrahimi, N. (1996). How to measure uncertainty in the residual life time distribution,  Sankhyā: The Indian Journal of Statistics, Series A, pages 48--56.
Rao, M., Chen, Y. and Vemuri, B.C. (2004). Cumulative residual entropy: a new measure of information,  IEEE Transactions on Information Theory,  50(6), 1220--1228.
 
Rao, M. (2005). More on a new concept of entropy and information,  Journal of Theoretical Probability,  18(4), 967--981.
Asadi, M. and Zohrevand, Y. (2007). On the dynamic cumulative residual entropy,  Journal of Statistical Planning and Inference,  137(6), 1931--1941.
Navarro, J., del Aguila, Y. and Asadi, M. (2010). Some new results on the cumulative residual entropy,  Journal of Statistical Planning and Inference,  140(1), 310--322.
Nair, N.U. and Preeth, M. (2009). On some properties of equilibrium distributions of order n,  Statistical Methods and Applications,  18(4), 453--464.
Nair, N.U. and Preeth, M. (2008). Multivariate equilibrium distributions of order n,  Statistics & Probability Letters,  78(18), 3312--3320.
Unnikrishnan Nair, N., Sunoj, S.M. and Rajesh, G. (2021). Cumulative residual entropy of equilibrium distribution of order n,  Communications in Statistics-Theory and Methods, pages 1--13.
Shannon, C.E. (1948). A mathematical theory of communication,  The Bell System Technical Journal,  27(3), 379--423.
Ebrahimi, N., Kirmani S.N.U.A. and Soofi E.S. (2007). Multivariate dynamic information,  Journal of Multivariate Analysis,  98(2), 328--349.
Arnold, B.C. (2014). Pareto distribution,  Wiley StatsRef: Statistics Reference Online, pages 1--10.
Rytgaard, M. (1990). Estimation in the Pareto distribution,  ASTIN Bulletin: The Journal of the IAA,  20(2), 201--216.
Kundu D. (2007). On hybrid censored Weibull distribution , Journal of Statistical Planning and Inference, 137(7),2127-2142.
Lindley,D.V.(1980) Approximate bayesian methods , Trabajos de estadística y de investigación operativa,31(1),223-245.
Arnold,B.C., Press,S.J.and others(1983) Bayesian inference for Pareto populations , Journal of Econometrics,21(3),287-306.
Kundu,D.and Joarder,A.(2006) Analysis of Type-II progressively hybrid censored data , Computational Statistics & Data Analysis,50(10),2509-2528.
Zaman,R.and Nasiri,P.(2019) Estimation of the Parameters of the Lomax Distribution using the EM Algorithm and Lindley Approximation , Mathematical Researches,4(2),201-210.
Lawless,J.F.(1982) Statistical models and methods for lifetime data Wiley ,New York.
Kundu,D.and Joarder,A.(2006) Analysis of Type-II progressively hybrid censored data , Computational Statistics & Data Analysis,50(10),2509-2528.
Greene, W.H. (2003). Econometric analysis. Pearson Education India.
Ma, Y. and Shi, Y. (2013). Inference for Lomax distribution based on type-II progressively hybrid censored data. Vidyasagar University, Midnapore, West-Bengal, India.
Renjini, K.R., Abdul Sathar, E.I. and Rajesh, G. (2018). Bayesian estimation of dynamic cumulative residual entropy for classical Pareto distribution,  American Journal of Mathematical and Management Sciences,  37(1), 1--13.
Renjini, K.R., Sathar, E.I.A. and Rajesh, G. (2016). Bayes estimation of dynamic cumulative residual entropy for Pareto distribution under type-II right censored data,  Applied Mathematical Modelling,  40(19-20), 8424--8434.
 
Yu, J., Gui, W. and Shan, Y. (2019). Statistical inference on the Shannon entropy of inverse Weibull distribution under the progressive first-failure censoring,  Entropy,  21(12), 1209.
Maiti, K., Kayal, S. and Kundu, D. (2022). Statistical Inference on the Shannon and Rényi Entropy Measures of Generalized Exponential Distribution Under the Progressive Censoring,  SN Computer Science,  3(4), 1--21.
Ahmadini, A.A.H., Hassan, A.S., Zaky, A.N. and Alshqaq, S.S. (2020). Bayesian inference of dynamic cumulative residual entropy from Pareto II distribution with application to COVID-19,  AIMS Math,  6(3), 2196--2216.
Shannon, C.E. (1948). A mathematical theory of communication,  The Bell System Technical Journal,  27(3), 379--423.
Harkness, W. and Shantaram, R. (1969). Convergence of a sequence of transformations of distribution functions,  Pacific Journal of Mathematics,  31(2), 403--415.
Gupta, R.C. (1979). On the characterization of survival distributions in reliability by properties of their renewal densities,  Communications in Statistics-Theory and Methods,  8(7), 685--697.
Pakes, A.G. (1996). Length biasing and laws equivalent to the log-normal,  Journal of Mathematical Analysis and Applications,  197(3), 825--854.
Nanda, A.K., Jain, K. and Singh, H. (1996). Properties of moments for s-order equilibrium distributions,  Journal of Applied Probability,  33(4), 1108--1111.
Pakes, A.G. and Navarro, J. (2007). Distributional characterizations through scaling relations,  Australian & New Zealand Journal of Statistics,  49(2), 115--135.
Nair N.U. and Sankaran P.G. (2010). Properties of a mean residual life function arising from renewal theory , Naval Research Logistics (NRL),57(4),373-379.
Lad F., Sanfilippo G., Agro G. (2015) Extropy: Complementary dual of entropy , Statistical Science,30(1),40-58.
Rao M., Chen Y., Vemuri B.C., Wang F.(2004) Cumulative residual entropy: a new measure of information , IEEE Transactions on Information Theory,50(6),1220-1228.
Jahanshahi S.M.A., Zarei H., Khammar A.H.(2020) On cumulative residual extropy , Probability in the Engineering and Informational Sciences,34(4),605-625.
Asadi M.and Zohrevand Y.(2007) On the dynamic cumulative residual entropy , Journal of Statistical Planning and Inference,137(6),1931-1941.
Muliere P., Parmigiani G.and Polson N.G.(1993) A note on the residual entropy function , Probability in the Engineering and Informational Sciences,7(3),413-420.
Qiu G.and Jia K.(2018) The residual extropy of order statistics , Statistics & Probability Letters,133,15-22.
Abdul Sathar E.I.and Nair R.Dhanya(2021) On dynamic survival extropy , Communications in Statistics-Theory and Methods,50(6),1295-1313.
Arnold B.C., Press S.J.and others(1983) Bayesian inference for Pareto populations , Journal of Econometrics,21(3),287-306.
Efron B.and Tibshirani R.(1986) Bootstrap Methods for Standard Errors ,Confidence Intervals,and Other Measures of , Statistical Science,1(1),54-75.
Balakrishnan N.and Sandhu R.A.(1995) A simple simulational algorithm for generating progressive Type-II censored samples , The American Statistician,49(2),229-230.
Journal of the Iranian Statistical Society
Volume 22, Issue 2
December 2023
Pages 119-146

  • Receive Date 22 June 2023
  • Revise Date 04 March 2024
  • Accept Date 03 July 2024