A Generalized Quasi-Likelihood Estimator for Nonstationary Stochastic Processes−Asymptotic Properties and Examples

2010 Mathematics Subject Classification: 62F12, 62M05, 62M09, 62M10, 60G42.

Let {Zn}n∈N be a real stochastic process on (Ω, F, Pθ0), where θ0 is a unknown p-dimensional parameter. We propose a GQLE (Generalized Quasi-Likelihood Estimator) of θ0 based on a single trajectory of the process and defined by ˆθn:=argminθ ∑k=1nΨk(Zk, θ), where Ψk(z, θ) is Fk-1-measurable, {Fn}n being an increasing sequence of σ-algebras. This class of estimators includes many different types of estimators such as conditional least squares estimators, least absolute deviation estimators and maximum likelihood estimators, and allows missing data, outliers, or infinite conditional variance. We give general conditions leading to the strong consistency and the asymptotic normality of ˆθn. The key tool is a uniform strong law of large numbers for martingales. We illustrate the results in the branching processes setting