The Efficiency and Consistency of Approximations to the Jackknife Variance Estimators.

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[sup -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 ((Multiple lines cannot be cannot be converted in ASCII text)) recomputations of a point estimate theta, which increases rapidly as n and d increase. Using techniques from survey sampling, a shortcut can be taken with theta evaluated only tn times, m ≪ (Multiple lines cannot be cannot be converted in ASCII text)). The efficiency and consistency of the resulting jackknife-sampling (hybrid) variance estimators (JSVE's) are studied. If m is chosen so that n/m arrow right 0, the increase in mean squared error by using the JSVE is relatively negligible. For the consistency of JSVE, m arrow right Infinity 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 theta needs large computations. The performance of JSVE is also studied in a simulation study. [ABSTRACT FROM AUTHOR]