Approximate Bayes Estimators of the Parameters of the Inverse Gaussian Distribution Under Different Loss Functions

  • Ilhan Usta Eskisehir Technical University, Department of Statistics, Eskisehir, Turkey
  • Merve Akdede Usak University Department of Statistics, Usak, Turkey
Keywords: Inverse Gaussian distribution, Bayes estimator, extension of Jeffrey’s prior, Lindley approximation, Tierney Kadane approximation


Inverse Gaussian is a popular distribution especially in the reliability and life time modelling, and thus the estimation of its unknown parameters has received considerable interest. This paper aims to obtain the Bayes estimators for the two parameters of the inverse Gaussian distribution under varied loss functions (squared error, general entropy and linear exponential). In Bayesian procedure, we consider commonly used non-informative priors such as the vague and Jeffrey’s priors, and also propose using the extension of Jeffrey’s prior. In the case where the two parameters are unknown, the Bayes estimators cannot be obtained in the closed-form. Hence, we employ two approximation methods, namely Lindley and Tierney Kadane (TK) approximations, to attain the Bayes estimates of the parameters. In this paper. the effects of considered loss functions, priors and approximation methods on Bayesian parameter estimation are also presented. The performance of Bayes estimates is compared with the corresponding classical estimates in terms of the bias and the relative efficiency throughout an extensive simulation study. The results of the comparison show that Bayes estimators obtained by TK method under linear exponential loss function using the proposed prior outperform the other estimators for estimating the parameters of inverse Gaussian distribution most of the time. Finally, a real data set is provided to illustrate the results.

Author Biographies

Ilhan Usta, Eskisehir Technical University, Department of Statistics, Eskisehir, Turkey

Ilhan Usta received his B.Sc. and B.Sc. (double major) degrees in Statistics and Mathematics from Anadolu University, Turkey in 2003 and 2004, respectively; M.Sc. and Ph.D. degrees in Statistics from Anadolu University, Institute of Science, Turkey in 2006 and 2009, respectively. He still works as a Professor at Eskisehir Technical University, Department of Statistics. His major areas of interest are theory of statistics, censored data and parameter estimation, Bayesian estimation, wind energy, entropy optimization distribution and portfolio theory.

Merve Akdede, Usak University Department of Statistics, Usak, Turkey

Merve Akdede received her B.Sc. degree in Statistics from Dokuz Eylul University, Turkey in 2008; M.Sc. degree in Statistics from Texas A&M University, USA in 2013. Although she has been a Ph.D. student at Eskisehir Technical University since 2015, she died suddenly at an early age in 2019. We will continue to keep her memory alive with this study and do good by her until the day we meet again.