Jackknifing

Jackknifing refers to a statistical analysis technique used to estimate the bias and variance of a statistical estimator. The method involves systematically recomputing the estimator by leaving out one observation at a time from the sample set. This process helps in understanding how changes in the data sample can affect the estimation. Jackknifing is particularly useful in situations where the sample size is small, and it provides a simple way to reduce the bias and variance of the estimator.

Overview

The concept of jackknifing was introduced by Maurice Quenouille in 1949 and was further developed by John Tukey in 1958. The primary goal of jackknifing is to improve the statistical properties of an estimator. In the process of jackknifing, each observation in a dataset is removed one at a time, and the statistic of interest is recalculated from the remaining observations. This procedure is repeated for each observation in the dataset. The jackknife estimates are then used to calculate the overall bias and variance of the estimator.

Procedure

The jackknifing procedure involves the following steps:

1. For a given dataset with n observations, calculate the statistic of interest using all n observations. This statistic is often denoted as θ.
2. Remove one observation from the dataset. The dataset now contains n-1 observations.
3. Calculate the statistic of interest using the remaining n-1 observations. This is done for each observation in the dataset, resulting in n jackknife estimates.
4. Use the n jackknife estimates to calculate the bias and variance of the original estimator θ.

Applications

Jackknifing is widely used in various fields of research, including biostatistics, economics, and ecology, among others. It is particularly useful in scenarios where the theoretical distribution of the estimator is unknown or difficult to derive. Jackknifing can also be applied to estimate the standard errors of the estimated parameters, making it a valuable tool for statistical inference.

Advantages and Limitations

One of the main advantages of jackknifing is its simplicity and ease of implementation. It does not require complex mathematical formulas or assumptions about the underlying distribution of the data. However, jackknifing has its limitations. It may not be effective for highly skewed distributions or when the data contains outliers. Additionally, jackknifing can be computationally intensive, especially for large datasets.

See Also

• Bootstrap (statistics)
• Cross-validation (statistics)
• Statistical inference