The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values \( \boldsymbol{x} = (x_1,x_2,\cdots,X_n) \). Let \( \boldsymbol{x}_i \) denote the vector $$ \boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n), $$
which equals the vector \( \boldsymbol{x} \) 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 \( \vec{X}_i \).