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A class of estimators for parameters of AR(1) processes obtained from previous nonparametric estimators. (Spanish. English summary) Zbl 0731.62139

Summary: Let \(\{X_ t\}_{t\in Z^+}\) be a stationary time series that follows the model: \(X_ t=\lambda +\rho X_{t-1}+e_ t\), where \(\{e_ t\}\) is a series of independent and identically distributed random variables with expectation zero and variance \(\sigma^ 2\). With an initial sample \(\{X_ 1,...,X_ n\}\) of the process we obtain in a first part nonparametric estimations \({\hat \tau}{}_ n\) and \({\hat \Omega}{}_ n\) of the prediction function \(\tau (x)=E[X_ t| X_{t-1}=x]\) and of the distribution function \(\Omega (x)=P\{X_ t\leq x\}\). With these estimations in a second part we define a family of parameter estimations of \(\{\lambda\),\(\rho\}\) that minimize the functional \[ {\hat \psi}(\lambda',\rho') =\int({\hat \tau}_ n(x)- \lambda'-\rho'x)^ 2 d{\hat \Omega}_ n(x). \] Consistency and asymptotic normality are obtained for such estimators.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G07 Density estimation
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References:

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