Bayesian estimation of renewal function for inverse Gaussian renewal process

Abstract

Two approximation methods are used to obtain the Bayes estimate for the renewal function of inverse Gaussian renewal process. Both approximations use a gamma-type conditional prior for the location parameter, a non-informative marginal prior for the shape parameter, and a squared error loss function. Simulations compare the accuracy of the estimators and indicate that the Tieney and Kadane (T–K)-based estimator out performs Maximum Likelihood (ML)- and Lindley (L)-based estimator. Computations for the T–K-based Bayes estimate employ the generalized Newton's method as well as a recent modified Newton's method with cubic convergence to maximize modified likelihood functions. The program is available from the author.

Keywords:

• Bayesian estimation
• inverse Gaussian
• Laplace's method
• modified Newton's method
• renewal function

Acknowledgements

The author is grateful to the referee and the associate editor for their constructive comments that improved the presentation of the article.