Generalized method of moments for an extended gamma process

Zeina Al Masry (1) , Sophie Mercier (2) , Ghislain Verdier (2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

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. 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 paper, 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 standard gamma processes against extended ones with a specific parametric shape function. Also, the performance of the proposed estimation methods is illustrated on simulated and real data.

Mots clés: Extended gamma process; Process with non-stationary independent increments; Parametric estimation; Laplace transform; Generalized method of moments

Domaines: Statistiques [stat]

Zeina Al Masry, Sophie Mercier, Ghislain Verdier. Generalized method of moments for an extended gamma process. Communications in Statistics - Theory and Methods, 2018, 47 (15), pp.3687-3714. ⟨10.1080/03610926.2017.1361988⟩. ⟨hal-01576969⟩