Jackknife resampling

Schematic of Jackknife Resampling

Jackknife resampling is a statistical technique used for estimating the bias and variance of a statistical estimate. The method involves systematically recomputing the statistical estimate leaving out one observation at a time from the sample set. This technique is particularly useful in situations where the sample size is small, and it provides insights into the distribution of the estimator itself. Jackknife resampling was proposed by Maurice Quenouille in 1949 and was further developed and popularized by John Tukey in the 1950s.

Overview

The basic idea behind jackknife resampling is to create multiple subsets of the original dataset by removing one observation at a time. For each subset, the statistic of interest is calculated. These statistics are then used to estimate the overall bias and variance of the estimator. The jackknife method is a precursor to the more general bootstrap resampling technique, which involves drawing samples with replacement to create multiple datasets.

Procedure

The procedure for jackknife resampling involves the following steps:

1. From a dataset of size n, create n new datasets of size n-1, each missing one observation from the original dataset.
2. Calculate the statistic of interest for each of these n datasets.
3. Aggregate these statistics to estimate the bias and variance of the original statistic.

Applications

Jackknife resampling is used in various fields including statistics, econometrics, and biostatistics. It is particularly useful for:

• Estimating the bias and variance of a sample statistic.
• Error estimation in statistical models.
• Validation of statistical models through resampling.

Advantages and Limitations

Advantages

• Simple to implement and understand.
• Requires no assumptions about the distribution of the data.
• Useful for small sample sizes.

Limitations

• Less efficient compared to the bootstrap method for larger datasets.
• Can be biased in certain situations, especially for non-smooth statistics.
• Not suitable for estimating the distribution of the statistic.

Comparison with Bootstrap Resampling

While both jackknife and bootstrap resampling are resampling methods used to estimate the distribution of a statistic, there are key differences:

• Bootstrap resampling involves sampling with replacement, creating samples that are the same size as the original dataset, while jackknife resampling systematically leaves out one observation at a time.
• Bootstrap is generally more flexible and applicable to a wider range of statistical estimates but requires larger sample sizes to be effective.

See Also

• Resampling (statistics)
• Cross-validation (statistics)
• Bootstrap resampling
• Variance
• Bias of an estimator