Modeling the Mileage Accumulation Process with Random Effects

Abstract

This article proposes a non stationary Gamma process with power-law shape function to model mileage accumulated processes in presence of heterogeneity in the sample paths. This heterogeneity is assumed to be charged to both the parameters of the shape function and is modeled through a bivariate distribution. Maximum likelihood estimates of the parameters indexing the bivariate distribution and of the scale parameter of the Gamma process, are obtained. The Monte Carlo sampling method is used to estimate the unconditional density function and the unconditional cumulative distribution function of the mileage accumulated by the population of vehicles up to a given age.

Keywords:

• Heterogeneity
• Maximum likelihood estimates
• Non stationary Gamma process
• Usage process

Mathematics Subject Classification:

• Primary 60J99
• Secondary 66K40

Acknowledgments

The author would like to thank the anonymous reviewers for their constructive comments and suggestions that improved the article. The present work has been partially developed with the contribution of PRIN-2008: “Innovation in service quality management: statistical approach and application in some fields of national interest” program of MIUR-Italy.