An Unbalanced Jackknife

Abstract

It is proved that the jackknife estimate $\tilde{\theta} = n\hat{\theta} - (n - 1)(\sum \hat{\theta}_{-i}/n)$ of a function $\theta = f(\beta)$ of the regression parameters in a general linear model $\mathbf{Y} = \mathbf{X\beta} + \mathbf{e}$ is asymptotically normally distributed under conditions that do not require $\mathbf{e}$ to be normally distributed. The jackknife is applied by deleting in succession each row of the $\mathbf{X}$ matrix and $\mathbf{Y}$ vector in order to compute $\hat{\mathbf{\beta}}_{-i}$, which is the least squares estimate with the $i$th row deleted, and $\hat{\theta}_{-i} = f(\hat\mathbf{\beta}_{-i})$. The standard error of the pseudo-values $\tilde{\theta}_i = n\hat{\theta} - (n - 1)\hat{\theta}_{-i}$ is a consistent estimate of the asymptotic standard deviation of $\tilde{\theta}$ under similar conditions. Generalizations and applications are discussed.