A Class of Non-Parametrically Constructed Parameter Estimators for a Stationary Autoregressive Model

Abstract

This article presents a new class of estimators for the parameters θ^t= (θ₁,…,θ_q) of the stationary autoregressive model ${\text{AR }}\left( {\text{q}} \right),{{\text{x}}_{\text{t}}}{\text{ = }}\mathop \Sigma \limits_{{\text{i = 1}}}^{\text{q}} {{\text{0}}_{\text{i}}}{{\text{x}}_{{\text{t - i}}}}{\text{ + }}{\varepsilon _{\text{t}}},$ with ${\text{E}}\left| {{{\text{X}}_{\text{t}}}} \right| = \,0,\,{\text{E}}\left[ {{\varepsilon _{\text{t}}}} \right] = 0$ and $\left( {{\varepsilon _t}} \right){\text{ }} = {\text{ }}{\sigma ^2}$. The new estimators are obtained by minimizing the functional

$$\hat \psi {\mkern 1mu} \left( \theta \right){\mkern 1mu} = \int {\left( {{{\hat \alpha }_n}{\mkern 1mu} \left( {\vec x} \right){\mkern 1mu} - {\mkern 1mu} {{\vec x}^t}{\mkern 1mu} \theta } \right)} {{\mkern 1mu} ^2}{\mkern 1mu} {\hat f_n}{\mkern 1mu} \left( {\vec x} \right){\mkern 1mu} d\vec x$$

where ${\hat \alpha _n}\,and\,{\hat f_n}$ are respectively non-parametric estimators of the prediction function $\alpha \left( {\vec x} \right){\text{ }} = {\text{ }}\alpha \left( {{x_1},...,{x_q}} \right){\text{ }} = {\text{ }}E\left[ {{X_t}/{X_{t{\text{ }} - {\text{ }}1}}{\text{ }} = {\text{ }}{x_1},...,{x_{t{\text{ }} - {\text{ }}q}}{\text{ }} = {\text{ }}{x_q}} \right]{\text{ }} = {\text{ }}{\theta ^t}\vec x$ and of f, the stationary initial density of (X₁,…,X_q). Consistency and asymptotic normality properties are proved.