The linear model with variance-covariance components and jackknife estimation

summary: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 $\vartheta$