An Advanced Statistical Method for Point Process Modelling With Missing Event Histories

Abstract

Researchers are always challenged in developing history-dependent point process models for recurrence events such as system failures when the early event history is missing. The raison d'être in most cases is the estimation of model parameters. Even for simple renewal point processes, the model estimation is difficult since the distribution of backward recurrence time is not explicitly given, except for the limiting distribution case. To fill the gap, this article establishes an effective statistical method for parameter estimation for history-dependent point process models with partial missing history. The proposed method addresses the missing history issue through a data augmentation (DA) technique integrated with a Markov Chain Monte Carlo (MCMC) simulation technique. The key novelty is the creation of an acceptance/rejection criterion to assure the validity of the augmented missing history. Next, the augmented and observed histories are combined to form a `complete' history which is in turn used for model estimation. The estimated parameters are then used to generate new missing history. An iterative procedure is implemented to warrant stationary distributions for the estimates after burn-in. The validity and efficiency of the proposed method are demonstrated using two simulation studies and one real-life case study of modelling municipal water pipe failures from different model estimation perspectives.

Keywords

• Bayesian inference
• data augmentation
• event history
• likelihood approach
• MCMC
• missing data