Maximum likelihood estimation for Gaussian process with nonlinear drift

Yuliya Mishura,Kostiantyn Ralchenko,Sergiy Shklyar

doi:10.15388/na.2018.1.9

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.

Highlights

Let B = {Bt, t 0} be a centered Gaussian process with known covariance function, B0 = 0
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 establish the strong consistency of both estimators

Summary

Introduction

We consider the model where B is the subfractional Brownian motion. Mention that similar problems for the model with linear drift driven by fractional Brownian motion were studied in [3, 9, 11, 15]. In [2, 16], the nonparametric functional estimation of the drift of a Gaussian processes was considered (such estimators for fractional and subfractional Brownian motions were studied in [8] and [19], respectively). Auxiliary results for nonrandom functions and integral equations are collected in the Appendix

The case of discrete-time observations

The likelihood function and MLE

The behaviour of the MLE for the increasing number of points

Assumptions on function G and process B

Consistency of the estimator

Model with subfractional Brownian motion