A comparative study on numerical, non-Bayes and Bayes estimation for the shape parameter of Kumaraswamy distribution

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq

2 Department of ways and Transportation, College of Engineering, Mustansiriyah University, Baghdad, Iraq

Abstract

This paper is considered with Kumaraswamy distribution. Numerical, non-Bayes and Bayes methods of estimation were used to estimate the unknown shape parameter. The maximum likelihood is obtained as a non-Bayes estimator. As well as, Bayes estimators under a symmetric loss function (De-groot and NLINEX) by using four types of informative priors three double priors and one single prior. In addition, numerical estimators are obtained by using Newton's method and the false position method. Simulation research is conducted for the comparison of the effectiveness of the proposed estimators. Matlab 2015 will be used to obtain the numerical results.

Keywords

[1] S.K. Abraheem, N.J.F. Al-Obedy and A.A. Mohammed, A comparative study on the double prior for reliability Kumaraswamy distribution with numerical solution, Baghdad Sci. J. 17(1) (2020) 159–165.
[2] A.K. Akbar, A. Mohammed, H. Zawar and T. Muhammad, Comparison of loss functions, for estimating the scale parameter of log-normal distribution using non-informative prior, Hacettepe J. Math. Stat. 45(6) (2016) 1831–1845.
[3] N.H. AL-NOOR and S.k. Ibraheem, On the maximum likelihood, Bayes and Empirical Bayes Estimation for the shape parameter, reliability and failure rate functions of Kumaraswamy distribution, Global J. Bio-Sci. Biotech. 5(1) (2016) 128–134.
[4] N.J.F. Al-Obedy, A.A. Mohammed and S.K. Abraheem, Numerical methods on the triple informative prior distribution for the failure rate basic Gompertz model, J. Univ. Anbar Pure Sci. 14(2) (2020) 88–94.
[5] S. S. AL Wan, Non-Bayes, Bayes and Empirical Bayes Estimator for Lomax Distribution, A Thesis Submitted to the Council of the College of Science at the AL-Mustansiriya University in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics, 2015.
[6] R. A. Bantan, F. Jamal, C. Chesneau and M. Elgrahy, Truncate inverted Kumaraswamy generated family of distributions with applications, Entropy 21(11) (2019) 1089.
[7] R. L. Burden and J. D. Faires, Numerical Analysis, Ninth Edition, Cengage Learning. Inc., Wadsworth Group, 2011.
[8] M. H. DeGroot, Optimal Statistical Decision, John Wiley & Sons, 2005.
[9] Y. Dodge, The Concise Encyclopedia of Statistics, Springer Science + Business Media, LLC., 2008.
[10] J. F. Epperson, Numerical Methods and Analysis, Second Edition, John Wiley and Sons, Inc. All Rights Reserved, 2013.
[11] R. Gholizadeh, A.M. Shirazi and S. Mosalmanzadeh, Classical and Bayesian estimation on the Kumaraswamy distribution using grouped and ungrouped data under difference loss functions, J. Appl. Sci. 11(12) (2011) 2154–2162.
[12] I. Ghosh, Bivariate and multivariate weighted Kumaraswamy distribution theory and applications, J. Stat. Theory Appl. 18(3) (2019) 198.
[13] W. Gautschi, Numerical Analysis, Second Edition, Springer Science+ Business Media, 2012.[14] A. Haq and M. Aslam, On the double prior selection for the parameter of Poisson distribution, Int. Stat. Statistics on the Internet, 2009 (November) http://Interstat.Statjounal.net
[15] A.F.M.S. Islam, M.K. Roy and M.M. Ali, A non-linear exponential (NLINEX) loss function in Bayesian analysis, J. Korean Data Inf. Sci. Soc. 15(4) (2004) 899–910.
[16] P. Kumaraswamy, Sinepower probability density function, J. Hydrol. 31 (1976) 181-184.
[17] P. Kumaraswamy, Extended sine power probability density function, J. Hydrol. 37 (1978) 81–89.
[18] S. F. Mohmmad and R. Batul, Bayesian estimation of shift point in shape parameter of inverse Gaussian distribution under different loss function, J. Optim. Indust. Engin. 18 (2015) 1–12.
[19] K.C. Patel and J.M. Patel, Analogical study of Newton-Raphson method & false position method, Int. J. Creative Res. Though. 8 (2020) 3508–3511.
[20] R.M. Patel and A.C. Patel, The double Prior selection for the parameter of exponential lifetime model under type II censoring, J. Modern Appl. Stat. Meth. 16(1) (2017) 406–427.
[21] S. Raja and S. P. Ahmad, Bayesian analysis of power function distribution under double Priors, J. Appl. Stat. 3(2) (2014) 239–249.
[22] M. Ronak, The double prior selection for the parameter of exponential lifetime model under type II censoring, JMASM 16(1) (2017) 406–427.
[23] A.F.M. Saiful Islam, Loss Functions, Utility Functions and Bayesian Sample Size Determination, A Thesis is Submitted for the Degree of Doctor of Philosophy in Queen Mary, University of London, 2011.
[24] S.G. Salman, Estimating the Parameter of Maxwell-Boltzman Distribution by Many Methods Employing Simulation, A Thesis Submitted to the Council of College Science for Women University of Baghdad as a Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics, 2017.
[25] A.K. Singh, R. Dalpatadu and A. Tsang, On estimation of parameters of the Pareto distribution, Actuarial Res. Clearing House 1 (1996) 407–409.
[26] S. V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, Second Edition, John Wiley and Sons Ltd., 2000.
Volume 13, Issue 1
March 2022
Pages 1417-1434
  • Receive Date: 07 October 2021
  • Accept Date: 29 November 2021