Simulation Study for Estimating the Parameters and Reliability Function of Weighted Exponential Distribution with Fuzzy Data
DOI:
https://doi.org/10.22401/93562w90Keywords:
Weighted Exponential Distribution , Maximum Likelihood Estimation , Bayesian Estimation , Newton-Raphson Algorithm , Lindley approximation , Fuzzy DataAbstract
This paper investigates the estimation of the two unknown parameters and the reliability function of the weighted exponential distribution. It explores Bayesian and non-Bayesian (maximum likelihood) estimation methods when the information available is in the form of fuzzy data. The Newton-Raphson algorithm is used to obtain the maximum likelihood estimates. In Bayes estimation, the symmetric squared error loss function is used. This loss function links equal importance to the losses due to overestimating and underestimating equal magnitude. Lindley approximation procedure in Bayesian estimation theory is used to evaluate the ratio of integrals. A comparative analysis using simulation is carried out to evaluate the performance of the obtained parameters estimators using mean squared error criteria and the performance of the obtained reliability estimators using integrated mean squared error criteria. The simulation results demonstrate that, for different sample sizes, the performance of Bayes estimates surpasses the maximum likelihood, and that all estimators perform consistently.
References
Gupta, R.D.; Kundu, D.; “A New Class of Weighted Exponential Distributions”. J. Theor. Appl. Stat., 43 (6): 621-634, 2009.
Azzalini, A.; “A Class of Distributions which Includes the Normal Ones”. Scand. J. Stat., 12 (2): 171–178, 1985.
Zamani, H.; Ismail, N.; “Poisson-Weighted Exponential Distribution and Its Application on Claim Count Data”. J. Qual. Meas. Anal., 6 (2): 57–65, 2010.
Makhdoom, I.; Nasiri, P.; “Estimation Parameters of Weighted Exponential Distribution with Presence of Outliers Generated from Exponential Distribution”. J. Appl. Sci., 11(23): 3811–3814, 2011.
Roy, S.; Adnan, M.A.S.; “Wrapped Weighted Exponential Distribution”. Stat. Probab. Lett., 82 (1): 77–83, 2012.
Abed Al-Kadim, K.; Hantoosh, A.F.; “Double Weighted Distribution and Double Weighted Exponential Distribution”. Math. Theory Model., 3 ( 5):124–134, 2013.
Khorram, E.; Farahani, Z.S.M., “Statistical Inference of Weighted Exponential Lifetimes Under Progressive Type-II Censoring Scheme”. Qual. Technol. Quant. Manage., 11(4): 433 – 451, 2014.
Badmus, N.I.; Bamiduro, A.T.; Rauf-Animasaun, M.A.; Akingbade, A.A.; “Beta Weighted Exponential Distribution: Theory and Application”. Int. J. Math. Anal. Optim: Theor. Appl., 1: 55–66, 2015.
Oguntunde, P.E.; Owoloko, E.A.; Balogun, O.S.; “On A New Weighted Exponential Distribution: Theory and Application”. Asian J. Appl. Sci., 9 (1):1 – 12, 2016.
Al-Noor, N.H.; Hussein, L.K.; “Weighted Exponential Distribution: Approximate Bayes Estimations with Fuzzy Data”. Al-Nahrain J. Sci., 174-185, 2018.
Al-Noor, N.H.; Hussein, L.K.; “Weighted Exponential-G Family of Probability Distributions”. Saudi. J. Eng. Technol., 3: 51-59, 2018.
Abd El-Bar, A.M.T.; Ragab I.E.; “On weighted exponential-Gompertz distribution: properties and application”. J. Taibah Univ. Sci., 13 (1): 616– 627, 2019.
Al-Noor, N.H.; Hussein, L.K.; “Kumaraswamy Weighted Exponential Distribution”. J. Arab Stat. Union, 5 (1): 28–36, 2020.
Mallick, A.; Ghosh, I.; Dey, S.; Devendra, K.; “Bounded Weighted Exponential Distribution with Applications”. Am. J. Math. Manage. Sci., 40 (1): 68–87, 2021.
Tiana, W.; Yang, Y.; “Change point analysis for weighted exponential distribution”. Commun. Stat. – Simul. Comput., 367–379, 2022.
Niu, R.; Tian, W.; Zhang, Y.; “Discriminating among Generalized Exponential, Weighted Exponential and Weibull Distributions”. Math., 11 (18), 3847, 2023.
Denoeux, T.; “Maximum Likelihood Estimation from Fuzzy Data Using the EM Algorithm”. Fuzzy. Sets Syst., 183(1): 72–91, 2011.
Zadeh, L.A.; “Probability Measures of Fuzzy Events”. J. Math.l Anal. Appl., 23 ( 2): 421–427, 1968.
Lee, J.C.; Lee, C.S., “An Approximate Maximum Likelihood Estimator in a Weighted Exponential Distribution”. J. Korean Data Inf. Sci. Soc., 23 (1): 219 – 225, 2012.
Mahdavi, A.; “Two Weighted Distributions Generated by Exponential Distribution”. J. Math. Ext., 9 (1): 1–12, 2015.
Shahbaz, S.; Shahbaz, M.Q.; Butt, N.S.; “A Class of Weighted Weibull Distribution”. Pakistan J. Stat. Oper. Res., 6 (1): 53–59, 2010.
Oguntunde, P.E.; “On the Exponentiated Weighted Exponential Distribution and Its Basic Statistical Properties”. Appl. Sci. Rep., 10 (3): 160–167, 2015.
Mahdy, M.M.R.; “A Class of Weighted Gamma Distribution and Its Properties”. Econ. Qual. Control, 26 : 133–144, 2011.
Saghir, A.; Saleem, M.; Khadim, A.; Tazeem, S.; “The Modified Double Weighted Exponential Distribution with Properties”. Int. J. Math. Comput., 1(1): 1–12, 2015.
Jain, K.; Singla, N.; Gupta, R.D.; “A Weighted Version of Gamma Distribution”. Probab. Stat., 34 : 89–111, 2014.
Ali, S.; Aslam, M.; Abbas, N.; Kazmi, S.; “Scale Parameter Estimation of the Laplace Model Using Different Asymmetric Loss Functions”. Int. J. Stat. Probab.,1(1): 105 –127, 2012.
Lindley, D.V.; “Approximate Bayesian Methods”. Trab. Estadistica, 31: 223–245, 1980.
Singh, S.K.; Singh, U.; Kumar, D.; “Bayesian Estimation of Parameters of Inverse Weibull Distribution”. Int. J. Stat. Probab., 40(7): 1597–1607, 2013.
Pak, A.; Parham, G.H.; Saraj, M.; “Inference for the Weibull Distribution Based on Fuzzy Data”. Rev. Colomb. Estadistica, 36 (2): 339–358, 2013.
Alqallaf, F.; Ghitany, M.E.; Agostinelli, C.; “Weighted Exponential Distribution: Different Methods of Estimations”. Appl. Math. Inf. Sci., 9 (3): 1167 – 1173, 2015.
Al-Noor, N.H.; AL-Sultany, Sh.A.K.; “Maximum Likelihood Estimates of Inverse Weibull Parameters and Reliability Function for Fuzzy Data under two Approximation Iterative Techniques”. Fif. Int. Sci. Conf. Arab Stat., 1-12, 2016.
Zadeh, L.A.; “Fuzzy Sets”. Inf. Control, 8: 338 – 353, 1965.
Divya, P.R.; “Quality Interval Acceptance Single Sampling Plan With Fuzzy Parameter Using Poisson Distribution”. Int. J. Adv. Res. Technol., 1: 115-125, 2012.
Piegat, A.; “A New Definition of the Fuzzy Set”. Int. J. Appl. Math. Comput. Sci., 15( 1): 125 – 140, 2005.
Sagban, M.H.; Fadhel, F.S.; “Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method”. Al-Nahrain J. Sci., 24 (4): 32-39, 2021.
Mohammed, R.A., Ahmed, B.A.; Fadhel, F.S.; “On fixed point theorem in fuzzy normed space”. Al-Nahrain J. Sci., 18(4): 138-143, 2015.
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