ONE-STEP JACKKNIFE FOR M-ESTIMATORS COMPUTED USING NEWTON’S METHOD

JUN SHAO

Department of Mathematics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5

(Received October 4, 1991; revised February 6, 1992)

Abstract. To estimate the dispersion of an M-estimator computed using Newton’s iterative method, the jackknife method usually requires to repeat the iterative process n times, where n is the sample size. To simplify the computation, one-step jackknife estimators, which require no iteration, are proposed in this paper. Asymptotic properties of the one-step jackknife estimators are obtained under some regularity conditions in the i.i.d. case and in a linear or nonlinear model. All the one-step jackknife estimators are shown to be asymptotically equivalent and they are also asymptotically equivalent to the original jackknife estimator. Hence one may use a dispersion estimator whose computation is the simplest. Finite sample properties of several one-step jackknife estimators are examined in a simulation study.

Key words and phrases: Asymptotic equivalence, asymptotic variance, computation of jackknife estimator, consistency, iteration, M-estimator, one-step estimator.