Maximum likelihood estimation for Gaussian process with nonlinear drift

Abstract

We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a known nonrandom function, and B is a centered Gaussian process. We construct the maximum likelihood estimators of the drift parameter θ based on discrete and continuous observations of the process X and prove their strong consistency. The results obtained generalize the paper [Yu. Mishura, K. Ralchenko, S. Shklyar, Maximum likelihood drift estimation for Gaussian process with stationary increments, Austrian J. Stat., 46(3–4): 67–78, 2017] in two directions: the drift may be nonlinear, and the noise may have nonstationary increments. As an example, the model with subfractional Brownian motion is considered.

Keywords

• Gaussian process
• discrete observations
• continuous observations
• maximum likelihood estimator
• strong consistency