On the Jackknife Statistic, Its Extensions, and Its Relation to $e_n$- Transformations

Abstract

In this paper, a complete overview is given of the theoretical development of various estimators generated by the jackknife statistic. In particular, the jackknife method is extended to stochastic processes by means of two estimators referred to as the $J_\infty$-estimator and the $J_\infty^{(2)}$-estimator. These estimators are studied in some detail and shown to have the same properties as the jackknife when one considers the length of the process record as the sample size. Finally, it is shown that the entire development of the jackknife procedures discussed in this paper can be considered as a direct parallel of earlier developments in numerical analysis surrounding the study of a transformation referred to as the $e_n$-transformation.