The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). The jackknife is a resampling method, we explained that this happens by scrambling the data in some way. When using the jackknife, this is done by systematically leaving out one observation from the vector of observed values \( \hat{x} = (x_1,x_2,\cdots,X_n) \). Let \( \hat{x}_i \) denote the vector
$$ \hat{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n), $$which equals the vector \( \hat{x} \) with the exception that observation number \( i \) is left out. Using this notation, define \( \widehat{\theta}_i \) to be the estimator \( \widehat{\theta} \) computed using \( \vec{X}_i \).