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The linear model with variance-covariance components and jackknife estimation
Applications of Mathematics, Tome 39 (1994) no. 2, pp. 111-125 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\theta^*$ be a biased estimate of the parameter $\vartheta$ based on all observations $x_1$, $\dots$, $x_n$ and let $\theta_{-i}^*$ ($i=1,2,\dots,n$) be the same estimate of the parameter $\vartheta$ obtained after deletion of the $i$-th observation. If the expectation of the estimators $\theta^*$ and $\theta_{-i}^*$ are expressed as $$ \align \mathrm{E}(\theta^*)=\vartheta+a(n)b(\vartheta) \\ \mathrm{E}(\theta_{-i}^*)=\vartheta+a(n-1)b(\vartheta)\qquad i=1,2,\dots,n, \endalign $$ where $a(n)$ is a known sequence of real numbers and $b(\vartheta)$ is a function of $\vartheta$, then this system of equations can be regarded as a linear model. The least squares method gives the generalized jackknife estimator. Using this method, it is possible to obtain the unbiased estimator of the parameter $\vartheta$.
Let $\theta^*$ be a biased estimate of the parameter $\vartheta$ based on all observations $x_1$, $\dots$, $x_n$ and let $\theta_{-i}^*$ ($i=1,2,\dots,n$) be the same estimate of the parameter $\vartheta$ obtained after deletion of the $i$-th observation. If the expectation of the estimators $\theta^*$ and $\theta_{-i}^*$ are expressed as $$ \align \mathrm{E}(\theta^*)=\vartheta+a(n)b(\vartheta) \\ \mathrm{E}(\theta_{-i}^*)=\vartheta+a(n-1)b(\vartheta)\qquad i=1,2,\dots,n, \endalign $$ where $a(n)$ is a known sequence of real numbers and $b(\vartheta)$ is a function of $\vartheta$, then this system of equations can be regarded as a linear model. The least squares method gives the generalized jackknife estimator. Using this method, it is possible to obtain the unbiased estimator of the parameter $\vartheta$.
DOI : 10.21136/AM.1994.134248
Classification : 62F10, 62J10
Keywords: Jackknife estimator; least squares estimator; linear model; estimator of variance-covariance components; consistency 
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Kudeláš, Jaromír. The linear model with variance-covariance components and jackknife estimation. Applications of Mathematics, Tome 39 (1994) no. 2, pp. 111-125. doi: 10.21136/AM.1994.134248

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