Statistical Inference for Multi-sample of Geometric Processes with Lognormal Distribution

Source abstract

The geometric process is an important counting process model that serves as an alternative to non-homogeneous Poisson process models. It is widely used in reliability analysis, particularly in the modeling of repairable systems, due to its monotonic behaviour. Accurate statistical inference for a geometric process is essential for appropriate data modeling. While extensive research has addressed inference in the single-sample case, real-world data are often collected from multiple sources, leading to multi-sample datasets. This introduces new challenges for statistical inference. In this study, we investigate the inference problem for a geometric process based on multi-sample data, under the assumption that the inter-arrival times of each process follow a lognormal distribution and that the samples are homogeneous. Maximum likelihood estimators of the model parameters are obtained, and their asymptotic characteristics are established. The Expectation-Maximization algorithm is adopted to compute the maximum likelihood estimators in case of censored observations. Additionally, we propose test statistics for assessing the homogeneity of the samples and for detecting the presence of a trend in the data. A simulation analysis is undertaken to evaluate and compare the maximum likelihood estimators against their non-parametric counterparts. Finally, the proposed inferential procedures are illustrated using a real dataset comprising failure data from two machining centers.