A perturbed gamma process with statistically dependent measurement errors

This paper proposes a new perturbed gamma process, where the distribution of the measurement errors is assumed to be statistically dependent on the level (or state) of the hidden process. The main distributional characteristics of the proposed model, such as the marginal and conditional probability distributions of the level of the hidden process, of the measured level, and of the measurement errors, are then provided. Other quantities of large interest, such as the “failed” and “false” alarm probabilities and the reliability functions, are also provided. The maximum likelihood estimate of the model parameters is discussed, both when the inspections are not destructive and when they are invasive or destructive. In case the inspections are destructive and the standard deviation of the measurement error depends linearly on the actual process level, some distributional approximations are also suggested in order to facilitate the parameters estimation. Finally, two numerical applications are used to illustrate the feasibility of the proposed perturbed model in an applicative framework.