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Estimation of Parameters for the Gumbel Type-I Distribution under Type-II Censoring Scheme


  • Asuman Yılmaz Department of Econometrics, Faculty of Economics and Administrative Sciences, Van Yüzüncü Yıl University, 65080 Van, Turkey.
  • Mahmut Kara Department of Econometrics, Faculty of Economics and Administrative Sciences, Van Yüzüncü Yıl University, 65080 Van, Turkey.



Bayesian Methods, Gumbel Type-I Distribution, Simulation, Type -II Censoring


This paper aims to decide the best parameter estimation methods for the parameters of the Gumbel type-I distribution under the type-II censorship scheme. For this purpose, classical and Bayesian parameter estimation procedures are considered. The maximum likelihood estimators are used for the classical parameter estimation procedure. The asymptotic distributions of these estimators are also derived. It is not possible to obtain explicit solutions of Bayesian estimators. Therefore, Markov Chain Monte Carlo, and Lindley techniques are taken into account to estimate the unknown parameters. In Bayesian analysis, it is very important to determine an appropriate combination of a prior distribution and a loss function. Therefore, two different prior distributions are used. Also, the Bayesian estimators concerning the parameters of interest under various loss functions are investigated. The Gibbs sampling algorithm is used to construct the Bayesian credible intervals. Then, the efficiencies of the maximum likelihood estimators are compared with Bayesian estimators via an extensive Monte Carlo simulation study. It has been shown that the Bayesian estimators are considerably more efficient than the maximum likelihood estimators. Finally, a real-life example is also presented for application purposes.


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Gumbel EJ. The return period of flood flows. Ann Math Statist. 1941; 12(2): 163-190.

Sarkar A, Deep S, Datta D, Vijaywargiya A, Roy R. Phanikanth VS. Weibull and Generalized Extreme Value Distributions for Wind Speed Data Analysis of Some Locations in India. KSCE J Civ Eng.. 2019; 23(8): 3476-3492.

Gómez Y, Bolfarine MH, Gómez HW. Gumbel distribution with heavy tails and applications to environmental data. Math Comput Simul. 2019; 157(C): 115-129.

Kiyani S, Kiyani V, Behdarvand N. Forecasting occur probability intense storm using Gumbel distribution; Case study: Nahavand township. Cent Asian J Environ Sci Technol Innov. 2021; 2(6): 219-226.

Abbas K, Fu J, Tang Y. Bayesian estimation of Gumbel type-II distribution. Data Sci J. 2013; 12(2013): 33-46.

Malinowska I, Szynal D. On characterization of certain distributions of kth lower (upper) record values. Appl Math Comput. 2008; 202(1): 338–347.

Yılmaz A, Kara M, Özdemir O. Comparison of different estimation methods for extreme value distribution. J Appl Stat. 2021; 48(13-15): 2259-2284.

Saleh HHH. Estimating Parametersof Gumbel Distribution For Maximum Values By using Simulation. Baghdad Sci J. 2016; 13(4): 0707.

Reyad H, Ahmed SO. E-Bayesian analysis of the Gumbel type-II distribution under type-II censored scheme. .Int J adv math Sci. 2015; 3(2): 108–120.

Saad AH, Ashour SK, Darwish DR. The Gumbel- Pareto Distribution: Theory and Applications. Baghdad Sci J. 2019; 16(4): 0937.

Kundu D, Raqab M Z. Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution. J Stat Plan Inference, 2012; 142(1), 41-47.

Altindağ Ö, Cankaya MN, Yalçinkaya A, Aydoğdu H. Statistical inference for the burr type III distribution under type II censored data. Commun Sci Univ Ank Series A1. 2017; 66(2): 297-310.

Nassar M, Okasha H, Albassam M. E‐Bayesian estimation and associated properties of simple step–stress model for exponential distribution based on type‐II censoring. Qual Reliab Eng Int. 2021; 37(3): 997-1016.

Okasha HM, El-Baz AH, Basheer AM. On Marshall-Olkin Extended Inverse Weibull Distribution: Properties and Estimation Using Type-II Censoring Data. J Stat Appl Pro Lett. 2020; 7(1): 9-21.

Xin H, Zhun J, Sun J, Zheng C, Tsai TR. Reliability inference based on the three-parameter Burr type XII distribution with type II censoring. Int J Reliab Qual. 2018; 25(02): 1850010.

Helu A, Samawi H. The inverse Weibull distribution as a failure model under various loss functions and based on progressive first-failure censored data. Qual Technol Quant Manag. 2015; 12(4): 517-535.

Awad M, Rashed H. Bayesian and Non - Bayesian Inference for Shape Parameter and Reliability Function of Basic Gompertz Distribution. Baghdad Sci J. 2020; 17(3): 0854.

Al-Baldawi TH. Comparison of maximum likelihood and some Bayes estimators for Maxwell distribution based on non-informative priors. Baghdad Sci J. 2013; 10(2): 480-488.

Yılmaz A, Kara M, Aydoğdu H. A study on comparisons of Bayesian and classical parameter estimation methods for the two-parameter Weibull distribution. Commun Fac Sci Univ Ank. Ser A1 Math Stat. 2020; 69(1): 576-602.

Lindley DV. Approximate Bayesian methods. 1980; 31(1): 223-237.

Smith AF, Roberts GO. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Series B Stat Methodol. 1993; 55(1): 3-23.

Metropolis N, Rosenbluth AW, Rosenbluth MN. Equation of state calculations by fast computing machines. J Chem Phys. 1953; 21(6): 1087-1092.

Hastings WK. Monte Carlo sampling methods using Markov chains and their applications. Biometrika. 1970; 57(1): 97-109.

Chen MH, Shao QM. Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat. 1999; 8(1): 69-92.

Dodson B. The Weibull Analysis Handbook. 2nd ed. Milwaukee, Wis.: ASQ Quality Press; 2006. xv+167 p. Available from:

Balakrishnan N, Kateri M. On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data. Stat Probab Lett. 2008; 78(17): 2971-2975.