An Unbalanced Jackknife

It is proved that the jackknife estimate θ̃ = nθ̂ - (n - 1)(∑ θ̂₋ᵢ/n) of a function θ = f(β) of the regression parameters in a general linear model Y = Xβ + e is asymptotically normally distributed under conditions that do not require e to be normally distributed. The jackknife is applied by deleting in succession each row of the X matrix and Y vector in order to compute β̂₋ᵢ, which is the least squares estimate with the ith row deleted, and θ̂₋ᵢ = f(β₋ᵢ). The standard error of the pseudo-values θ̃ᵢ = nθ̂ - (n - 1)θ̂₋ᵢ is a consistent estimate of the asymptotic standard deviation of θ̃ under similar conditions. Generalizations and applications are discussed.