Third‐Order Asymptotic Properties Of Estimators In Gaussian Arma Processes With Unknown Mean

. This paper deals with the third‐order asymptotic theory for Gaussian autoregressive moving‐average (ARMA) processes with unknown mean μ. We are interested in the estimation of ρ= (α1…, αp, β1…, βq), where α1…, αρ and β1…, βq are the coefficients of the autoregressive part and the moving‐average part, respectively. First, we investigate the third‐order asymptotic optimality of the bias adjusted maximum likelihood estimator (MLE) of ρ in the presence of the nuisance parameters μ and s̀2 (innovation variance). Next, for a Gaussian AR(1μμ, s̀2), we propose a mean corrected estimator αc1c2 of the autoregressive coefficient. We make a comparison between the bias adjusted estimator αc1c2* and the bias adjusted MLE, in terms of their probabilities of concentration around the true value, or equivalently, in terms of their mean squared errors. Finally some numerical studies are provided in order to verify the third‐order asymptotic theory.