## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 5 (1974), 880-891.

### An Unbalanced Jackknife

#### Abstract

It is proved that the jackknife estimate $\tilde{\theta} = n\hat{\theta} - (n - 1)(\sum \hat{\theta}_{-i}/n)$ of a function $\theta = f(\beta)$ of the regression parameters in a general linear model $\mathbf{Y} = \mathbf{X\beta} + \mathbf{e}$ is asymptotically normally distributed under conditions that do not require $\mathbf{e}$ to be normally distributed. The jackknife is applied by deleting in succession each row of the $\mathbf{X}$ matrix and $\mathbf{Y}$ vector in order to compute $\hat{\mathbf{\beta}}_{-i}$, which is the least squares estimate with the $i$th row deleted, and $\hat{\theta}_{-i} = f(\hat\mathbf{\beta}_{-i})$. The standard error of the pseudo-values $\tilde{\theta}_i = n\hat{\theta} - (n - 1)\hat{\theta}_{-i}$ is a consistent estimate of the asymptotic standard deviation of $\tilde{\theta}$ under similar conditions. Generalizations and applications are discussed.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 5 (1974), 880-891.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342811

**Digital Object Identifier**

doi:10.1214/aos/1176342811

**Mathematical Reviews number (MathSciNet)**

MR356353

**Zentralblatt MATH identifier**

0289.62042

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

15 35 Jackknife pseudo-value general linear model multiple regression asymptotic normality

#### Citation

Miller, Rupert G. An Unbalanced Jackknife. Ann. Statist. 2 (1974), no. 5, 880--891. doi:10.1214/aos/1176342811. https://projecteuclid.org/euclid.aos/1176342811