## The Annals of Statistics

### Bootstrap Methods: Another Look at the Jackknife

B. Efron

#### Abstract

We discuss the following problem: given a random sample $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ from an unknown probability distribution $F$, estimate the sampling distribution of some prespecified random variable $R(\mathbf{X}, F)$, on the basis of the observed data $\mathbf{x}$. (Standard jackknife theory gives an approximate mean and variance in the case $R(\mathbf{X}, F) = \theta(\hat{F}) - \theta(F), \theta$ some parameter of interest.) A general method, called the "bootstrap," is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.

#### Article information

Source
Ann. Statist. Volume 7, Number 1 (1979), 1-26.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176344552

JSTOR