Asymptotic Properties of the LS-estimator of a Gaussian Autoregressive Process by an Averaging Method

Abstract

This article deals with the problem of parameter estimation of a continuous-time p-dimensional Gaussian autoregressive process. In the stable case, we combine averaging and weighting methods to establish, for the weighted LS-estimator (least-squares estimator)˜ of , an almost-sure central limit theorem (ASCLT), a quadratic strong law (QSL) and a central limit theorem (CLT) associated to the QSL with arithmetic convergence rates. In the unstable case, we establish for the LS-estimatorˆ of , the same optimal asymptotic properties with arithmetic convergence rates. In both cases, using the QSL property, we construct a new estimator of the noise covariance 2 and we specify its weak convergence rates.

Asymptotic properties of AD(1, n) model and its maximum likelihood estimator

“…Further, stationarity and ergodicity studies for affine diffusion processes found a particular interest by many authors, see, e.g., [4,10,18,19,25,26,27] and [43], where they give sufficient conditions for the existence of a unique stationary distribution and for the ergodic property. Moreover, several papers have studied the drift parameter estimation in different models, see, e.g., [2,3,5,6,7,9,15] and the references therein. It should be noted that statistical inferences of AD(1, n) model were not considered and more generally, inferences of affine multidimensional diffusions are rarely treated.…”