Estimation and Prediction for the Bounded Transformed Gamma Process: A Bayesian Approach

Abstract

In this article, an informative Bayesian approach is proposed for the bounded transformed gamma process, a novel stochastic process recently proposed in the literature to describe bounded above, monotonic increasing, degradation phenomena. The proposed approach is used to analyze a set of real wear data of the cylinder liners of a Diesel engine. Several scenarios, which differ in terms of the quality of the available prior knowledge, are considered and suitable prior distributions are suggested for each of them. In addition, detailed instructions are provided to help potential users incorporate into the suggested prior distributions all and solely the pieces of prior information that are available and sound. In particular, weak prior distributions are also suggested for situations in which available information is poor and/or there is no prior information to exploit. The proposed approach is used to estimate the process parameters and some functions thereof, such as the mean degradation level, the residual reliability of a unit, and to predict the future degradation growth and the useful lifetime. Point estimation and prediction under the (asymmetric) general entropy loss function are also performed to properly deal with situations where overestimation is costlier than underestimation, or vice versa. Estimates and predictions are computed by using proper Markov Chain Monte Carlo algorithms. Results obtained by analyzing wear data of the liners are compared both with those provided by classical methods and with those obtained by using Bayesian approaches based on vague priors. Finally, a sensitivity analysis is developed to study the impact of different prior distributions on the estimates of the parameters.