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Zbl 0605.62066
Toyooka, Yasuyuki
Second-order risk structure of GLSE and MLE in a regression with a linear process. (English)
[J] Ann. Stat. 14, 1214-1225 (1986). ISSN 0090-5364

The paper deals with estimation of $\beta \in R\sp p$ in the model $$ y\sb t=x\sb t'\beta +\sum\sp{\infty}\sb{j=0}g\sb j(\theta)\epsilon\sb{t- j}, $$ where $\{x\sb t\}$ is fixed, and $\theta \in R\sp 1$ is unknown. To obtain a GLSE, $\theta$ is estimated by the minimizing value ${\hat \theta}$ of a Whittle functional for $\tilde u=y-X{\hat \beta}$, ${\hat \beta}$ being the LSE for $\beta$. With the covariance matrix V($\theta)$ of the linear error-process $\{u\sb t\}$, the GLSE is $$ {\hat \beta}\sb{\hat w}=\{X'V\sp{-1}({\hat \theta})X\}\sp{-1}X'V\sp{- 1}(\theta)y. $$ ${\hat \beta}\sb{\hat w}$ is shown to be unbiased and the leading term of the estimation effect of $\theta$ on the covariance matrix of ${\hat \beta}\sb{\hat w}$ is evaluated.
[R.Schlittgen]

MSC 2000:: *62J05 Linear regression
62J10 Analysis of variance, etc.

Keywords: second-order expansion of the risk matrix; Grenander's condition; generalized least squares estimator; maximum likelihood estimator; GLSE; Whittle functional; linear error-process

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