The **jackknife **(“leave one out”) can be used to reduce bias and estimate standard errors. It is an alternative to the bootstrap method.

## Comparison to Bootstrap

Like the bootstrap, the Jackknife involves resampling. The main differences are:

- The bootstrap involves sampling with replacement, while the jackknife involves sampling without replacement.
- The bootstrap tends to be more computationally intensive.
- The jackknife’s estimated standard error tends to be larger than bootstrap resampling.

## Overview of the Jackknife Procedure

The basic idea is to calculate the estimator (e.g. the sample mean) by sequentially deleting a single observation from the sample. The estimator is recomputed until there are *n* estimates for a sample size of *n*. As a simple example, let’s say you had five data points X_{1}, X_{2}, X_{3}, X_{4}, X_{5}. You would calculate the estimator five times, for:

- X
_{1}, X_{2}, X_{3}, X_{4}, X_{5}. - X
_{2}, X_{3}, X_{4}, X_{5}. - X
_{3}, X_{4}, X_{5}. - X
_{4}, X_{5}. - X
_{5}.

Once you have your *n* estimates

,

the standard error is calculated with the following formula:

## Why Use Jackknife Estimation?

Jackknife estimation is usually used when it’s **difficult or impossible to get estimators using another method.** For example:

- No theoretical basis is available for estimation,
- The statistics’s function is challenging to work with (e.g. a function with no closed form integral, which would make the usual method (the delta method) impossible),

For large samples, the Jackknife method is roughly equivalent to the delta method.

## References

The Bootstrap and Jack knife. Retrieved November 2, 2019 from: https://www.biostat.washington.edu/sites/default/files/modules/2017_sisg_1_9_v3.pdf

McIntosh, A. The Jack knife Estimation Method. Retrieved November 2, 2019 from: http://people.bu.edu/aimcinto/jackknife.pdf

Ramachandran, K. & Tsokos, C. (2014). Mathematical Statistics with Applications in R. Elsevier.

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