Non-Parametric Inference for Multi-Sample of Geometric Processes with Application to Multi-System Repair Process Modeling

The geometric process is a significant monotonic stochastic process widely used in the fields of applied probability, particularly in the failure analysis of repairable systems. For repairable systems modeled by a geometric process, accurate estimation of model parameters is essential. The inference problem for geometric processes has been well-studied in the case of single-sample data. However, multi-sample data may arise when the repair processes of multiple systems are observed simultaneously. This study addresses the non-parametric inference problem for geometric processes based on multi-sample data. Several non-parametric estimators are proposed using the linear regression method, and their asymptotic properties are established. In addition, test statistics are introduced to assess sample homogeneity and to evaluate the significance of the trend observed in the process. The performance of the proposed estimators is evaluated through a comprehensive simulation study under small-sample settings. An artificial data analysis is conducted to model the repair processes of multiple repairable systems using the geometric process. Furthermore, a real-world dataset consisting of multi-sample failure data from two shared memory processors of the Blue Mountain supercomputer is analyzed to demonstrate the practical applicability of the method in multi-sample failure data analysis.