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

W. González Manteiga and J. M. Vilar Fernández
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W. González Manteiga: Universidad de Santiago de Compostela, Departamento de Estadística e Investigación Operativa Facultad de Matemáticas
J. M. Vilar Fernández: Universidad de Santiago de Compostela, Departamento de Estadística e Investigación Operativa Facultad de Matemáticas

A chapter in Mathematical Statistics and Probability Theory, 1987, pp 85-95 from Springer

Abstract: Abstract This article presents a new class of estimators for the parameters θt= (θ1,…,θq) 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,…,Xq). Consistency and asymptotic normality properties are proved.

Keywords: Prediction Function; Bound Convergence Theorem; Linear Regression Estimate; Kernel Regression Estimation; Nonparametric Time Series (search for similar items in EconPapers)
Date: 1987
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-3965-3_9

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DOI: 10.1007/978-94-009-3965-3_9

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