Estimation of a Shape Parameter of a Gompertz-lindley Distribution Using Bayesian and Maximum Likelihood Methods
Innocent Boyle Eraikhuemen *
Department of Physical Sciences, Benson Idahosa University Benin-City, Nigeria.
Abraham Iorkaa Asongo
Department of Statistics and Operations Research, MAUTech, P. M. B. 2076, Yola, Nigeria.
Adamu Abubakar Umar
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia and Department of Statistics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria-Kaduna, Nigeria.
Isa Abubakar Ibrahim
Department of Mathematics and Computer Science, Federal University Kashere, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The Gompertz-Lindley distribution is an extension of the Lindley distribution with three parameters. It was found to be more flexible for modeling real life events. The distribution contains two shape parameters and a scale parameter. Despite the necessity of parameter estimation theory in modeling, it has not been shown that a method of estimation method is better for any of these three parameters of the Gompertz-Lindley distribution. This paper identifies the best estimation method for the shape parameter of the Gompertz-Lindley distribution by deriving Bayesian estimators for the shape parameter of the distribution using two non-informative prior distributions (Uniform and Jeffery) and an informative prior (gamma) under squared error loss function (SELF), quadratic loss function (QLF) and precautionary loss function (PLF). These estimators were evaluated and the results compared with the maximum likelihood estimation method using Monte Carlo simulations with the mean square error (MSE) as a criterion for choosing the best estimator.
Keywords: Gomperz-Lindley distribution, bayesian analysis, prior distributions, loss functions, maximum likelihood estimation and mean square error
How to Cite
References
Lindley DV. Fiducial distributions and Bayes Theorem. J. Roy. Stat. Soc. Series B (Methodological). 1958;102–107.
Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its application. Math. Comput. Simul. 2008;78 (4):493–506.
Mazucheli J, Achcar JA. The Lindley distribution applied to competing risks lifetime data. Comput. Methods Programs Biomed. 2011;104:188–192.
Krishna H, Kumar K. Reliability estimation in Lindley distribution with progressively type-ii right censored sample. Math. Comput. Simul. 2011;82:281–294.
Singh B, Gupta PK. Load-sharing system model and its application to the real data set. Math. Comput. Simul. 2012;82:1615–1629.
Al-Mutairi DK, Ghitany ME, Kundu D. Inferences on stress-strength reliability from Lindley distributions. Commun. Stat. - Theory and Methods. 2013;42:1443–1463.
Sharma VK, Singh SK, Singh U. A new upside-down bathtub shaped hazard rate model for survival data analysis. Appl. Math. Comput. 2014;239:242–253.
Merovci F. Transmuted Lindley distribution. Int. J. Open Problems Compt. Math. 2013;6(2):63-74.
Ashour SK, Eltehiwy MA. Exponentiated power Lindley distribution. J. Adv. Res. 2015;6:895–905.
Nadarajah S, Bakouch HS, Tahmasbi R. A generalized Lindley distribution. Sankhya B. 2011;73:331–359.
Elgarhy MM, Rashed M, Shawki AW. Transmuted Generalized Lindley Distribution. International Journal of Mathematics Trends and Technology. 2016;29(2):145-154.
Alkarni SH. Extended Power Lindley Distribution: A New Statistical Model for Non-Monotone Survival Data. European Journal of Statistics and Probability. 2015;3(3):19-34.
Al-khazaleh M, Al-Omari AI, Al-khazaleh AMH. Transmuted Two-Parameter Lindley Distribution. Journal of Statistics Applications & Probability. 2016;5(3):421-432.
Shanker R, Shukla KK, Shanker R, Leonida TA. A Three-Parameter Lindley Distribution. American Journal of Mathematics and Statistics. 2017;7(1):15-26.
Ghitany M, Al Mutairi D, Balakrishnan N, and Al Enezi I. Power Lindley distribution and associated inference. Computational Statistics and Data Analysis. 2013;64:20-33.
Merovci F, Elbatal I. Transmuted Lindley-geometric distribution and its applications. J. Stat. Appl. Probability Lett. 2014;3:77–91.
Merovci F, Sharma VK. The beta Lindley distribution: properties and applications. J. Appl. Math. 2014;1–10.
Akmakyapan S, Kadlar GZ. A new customer lifetime duration distribution: the Kumaraswamy Lindley distribution. Int. J. Trade, Economics Finance. 2014;5:441–444.
Koleoso PO, Chukwu AU, Bamiduro TA. A three-parameter Gompertz-Lindley distribution: Its properties and applications. J. of Math. Theo. & Mod. 2019;9(4):29-42.
Cordeiro GM, Afify AZ, Ortega EMM, Suzuki AK & Mead ME. The odd Lomax generator of distributions: Properties, estimation and applications. Journal of Computational and Applied Mathematics. 2019;347:222–237.
Cordeiro GM, Ortega EMM, Popovic BV & Pescim RR. The Lomax generator of distributions: Properties, minification process and regression model. Applied Mathematics and Computation. 2014;247:465-486.
Afify MZ, Yousof HM, Cordeiro GM, Ortega EMM. & Nofal ZM. The Weibull Frechet Distribution and Its Applications. Journal of Applied Statistics. 2016;1-22.
Tahir MH, Zubair M, Mansoor M, Cordeiro GM & Alizadeh M. A New Weibull-G family of distributions. Hacettepe Journal of Mathematics and Statistics. 2016;45(2):629-647.
Ieren TG, Yahaya A. The Weimal Distribution: its properties and applications. Journal of the Nigeria Association of Mathematical Physics. 2017;39:135-148.
Bourguignon M, Silva RB, Cordeiro GM. The weibull-G family of probability distributions. Journal of Data Science. 2014;12:53-68.
Gomes-Silva F, Percontini A, De Brito E, Ramos MW, Venancio R & Cordeiro GM. The Odd Lindley-G Family of Distributions. Austrian Journal of Statistics. 2017;46:65-87.
Ieren TG, Koleoso PO, Chama AF, Eraikhuemen IB & Yakubu N. A Lomax-inverse Lindley Distribution: Model, Properties and Applications to Lifetime Data. Journal of Advances in Mathematics and Computer Science. 2019;34(3-4): 1-28.
Alzaatreh A, Famoye F, Lee C. A new method for generating families of continuous distributions. Metron. 2013;71: 63–79.
Ieren TG, Kromtit FM, Agbor BU, Eraikhuemen IB & Koleoso PO. A Power Gompertz Distribution: Model, Properties and Application to Bladder Cancer Data. Asian Research Journal of Mathematics. 2019;15(2):1-14.
Ieren TG, Oyamakin SO & Chukwu AU. Modeling Lifetime Data With Weibull-Lindley Distribution. Biometrics and Biostatistics International Journal. 2018; 7(6):532‒544.
Martz HF & Waller RA. Bayesian reliability analysis. New York, NY: John Wiley; 1982.
Ibrahim JG, Chen MH & Sinha D. Bayesian survival analy. New York, NY: Springer-Verlag; 2001.
Singpurwalla ND. Reliability and risk: A Bayesian perspective. Chichester: John Wiley; 2006.
Dey S. Bayesian estimation of the shape parameter of the generalized exponential distribution under different loss functions. Pakistan Journal of Statistics and Operation Research. 2010;6(2): 163-174.
Ahmed AOM, Ibrahim NA, Arasan J, Adam MB. Extension of Jeffreys’ prior estimate for weibull censored data using Lindley’s approximation. Austr. J. of B. and Appl. Sci. 2011;5(12):884–889.
Pandey BN, Dwividi N, Pulastya B. Comparison between Bayesian and maximum likelihood estimation of the scale parameter in Weibull distribution with known shape under linex loss function, J. of Sci. Resr. 2011;55:163–172.
Mabur TM, Omale A, Lawal A, Dewu MM & Mohammed S. Bayesian Estimation of the Scale Parameter of the Weimal Distribution. Asian Journal of Probability and Statistics. 2018;2(4):1-9.
Ahmad K, Ahmad SP, Ahmed A. On parameter estimation of erlang distribution using bayesian method under different loss functions, Proc. of Int. Conf. on Adv. in Computers, Com., and Electronic Eng. 2015;200–206.
Ieren TG, Chama AF, Bamigbala OA, Joel J, Kromtit FM, Eraikhuemen IB. On A Shape Parameter of Gompertz Inverse Exponential Distribution Using Classical and Non Classical Methods of Estimation. Journal of Scientific Research & Reports. 2020;25(6):1-10.
Eraikhuemen IB, Bamigbala OA, Magaji UA, Yakura BS, Manju KA. Bayesian Analysis of Weibull-Lindley Distribution Using Different Loss Functions. Asian Journal of Advanced Research and Reports. 2020;8(4):28-41.
Ieren TG, Oguntunde PE. A Comparison between Maximum Likelihood and Bayesian Estimation Methods for a Shape Parameter of the Weibull-Exponential Distribution. Asian J. of Prob. and Stat. 2018;1(1):1-12.
Ieren TG, Chukwu AU. Bayesian Estimation of a Shape Parameter of the Weibull-Frechet Distribution. Asian Journal of Probability and Statistics. 2018;2(1):1-19.
Krishna H, Goel N. Maximum Likelihood and Bayes Estimation in Randomly Censored Geometric Distribution. Journal of Probability and Statistics. 2017;12.
Gupta PK & Singh AK. Classical and Bayesian estimation of Weibull distribution in presence of outliers. Cogent Math. 2017;4:1300975.
Gupta I. Estimation of Parameter And Reliability Function of Exponentiated Inverted Weibull Distribution using Classical and Bayesian Approach. Int. J. of Recent Sci. Res. 2017;8(7):18117-18819.
Ahmad K, Ahmad SP & Ahmed A. Classical and Bayesian Approach in Estimation of Scale Parameter of Nakagami Distribution. J. of Prob. and Stat., Article ID 7581918, 8;2016. Available:http://dx.doi.org/10.1155/2016/7581918.
Preda V, Eugenia P & Alina C. Bayes Estimators of Modified-Weibull Distribution parameters using Lindley's approximation. Wseas Transactions on Mathematics. 2010;9(7):539-549.
Abbas K, Abbasi NY, Ali A, Khan SA, Manzoor S, Khalil A, Khalil U, Khan DM, Hussain Z & Altaf M. Bayesian Analysis of Three-Parameter Frechet Distribution with Medical Applications. Computational and Mathematical Methods in Medicine. 2019, Article ID 9089856, 8 Available:https://doi.org/10.1155/2019/9089856.
Ramos PL, Louzada F, Ramos E. Posterior Properties of the Nakagami-m Distribution Using Noninformative Priors and Applications in Reliability. IEEE Transactions on reliability. 2017;67(1):105-117.
Ramos PL, Louzada F. Bayesian reference analysis for the Generalized Gamma distribution. IEEE Communications Letters. 2018;22(9):1950-1953.
Tomazella VLD, de Jesus SR, Louzada F, Nadarajah S, Ramos PL. Reference Bayesian analysis for the generalized lognormal distribution with application to survival data. Statistics and Its Interface, 2020;13(1):139-149.
Azam Z, Ahmad SA. Bayesian Approach in Estimation of Scale Parameter of Nakagami Distribution. Pakistan J. of Stat. and Operat. Res. 2014;10(2): 217-228.
Norstrom JG. The use of precautionary loss functions in risk analysis. IEEE Trans. Reliab. 1996;45(3):400–403 Available:http://dx.doi.org/10.1109/24.536992.