Misspecification Analysis of a Gamma- with an Inverse Gaussian-Based Degradation Model in the presence of Measurement Error

Gamma- and inverse Gaussian-based degradation models are natural choices for modelling monotonic degradation phenomena. Although not equivalent, in many applications these models are treated to each other. This situation makes the model misspecification problem interesting and important, especially when data are affected by measurement error, because from this kind of data it is not possible to check whether the selected model is able to fit the hidden degradation process or not. Motivated by the above considerations, in the paper we have carried out a small (i.e., very preliminary) Monte Carlo study to evaluate the effect produced by a misspecification of a gammawith an inverse Gaussian-based perturbed degradation process. The study is performed considering as reference model a perturbed Gamma process recently proposed in the literature. The competing model is new and has been constructed with the aim of facilitating the misspecification study. In fact, it is an inverse Gaussian-based perturbed degradation model that share the same parameters and the same error term of the reference model. By virtue of the adopted setup, the mean and variance functions of the considered hidden Inverse Gaussian and Gamma processes have identical functional forms. The measurement error is modelled by a 3-parameter inverse gamma random variable, which depends in stochastic sense on the hidden degradation level. Model parameters are estimated by adopting the maximum likelihood method, via a sequential Monte Carlo approach. The fitting ability of the considered competing models is evaluated by using the Akaike information criterion. The effect of the misspecification is highlighted on the maximum likelihood estimate of mean remaining useful life. The impact of the presence of measurement error on the severity of the misspecification problem is also evaluated by comparing the obtained results with those attained by performing the same misspecification analysis in the absence of measurement error.