Generalized method of moments for an extended gamma process

ABSTRACT

In reliability theory, a widely used process to model the phenomena of the cumulative deterioration of a system over time is the standard gamma process (SGP). Based on several restrictions, such as a constant variance-to-mean ratio, this process is not always a suitable choice to describe the deterioration. A way to overcome these restrictions is to use an extended version of the gamma process introduced by Cinlar (1980), which is characterized by shape and scale functions. In this article, the aim is to propose statistical methods to estimate the unknown parameters of parametric forms of the shape and scale functions. We here develop two generalized methods of moments (Hansen 1982), based either on the moments or on the Laplace transform of an extended gamma process. Asymptotic properties are provided and a Wald-type test is derived, which allows to test SGPs against extended ones with a specific parametric shape function. Also, the performance of the proposed estimation methods is illustrated on simulated and real data.

KEYWORDS:

• Extended gamma process
• Generalized method of moments
• Laplace transform
• Parametric estimation
• Process with non stationary independent increments.

MATHEMATICS SUBJECT CLASSIFICATION:

• 60G51
• 62M05
• 62F12
• 60K10

Additional information

Funding

This work is part of Zeina Al Masry’s PhD studies which was supported by a French Ministry grant.