The Efficiency and Consistency of Approximations to the Jackknife Variance Estimators

Abstract

The problem considered is the computation reduction for general delete- d jackknife variance estimators. The delete- d jackknife estimator was proved consistent (Shao and Wu 1986), and in this article its mean squared error is shown to have order o ( n –2 ), where n is the sample size. These properties are not shared by the traditional delete-1 jackknife in some situations. Use of the delete- d jackknife, however, requires ( n d ) recomputations of a point estimate θ, which increases rapidly as n and d increase. Using techniques from survey sampling, a shortcut can be taken with θ evaluated only m times, m ≪ ( n d ). The efficiency and consistency of the resulting jackknife-sampling (hybrid) variance estimators (JSVE's) are studied. If m is chosen so that n/m → 0, the increase in mean squared error by using the JSVE is relatively negligible. For the consistency of JSVE, m → ∞ is sufficient. Hence the JSVE with m < n can also be used to alleviate the computational burden for the delete-1 jackknife in the case where n is large and evaluating θ needs large computations. The performance of JSVE is also studied in a simulation study.