Jackknife Resampling

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What Is Jackknife Resampling?

Jackknife resampling is a statistical technique for determining an estimate's standard error and bias using a re-computation of the estimate taken from sub-samples within provided data. It aims to provide estimates of standard error or bias through its simple resampling method.

Jackknife resampling is quite computationally intensive when used for the resampling of large samples. It is generated by sequentially removing single cases from the original data, known as delete-one jackknife. This method is suitable for single as well as multiple variables. It usually produces different results when repeated on the same sample. Thus, in Jackknife, one observes that one data point is removed one at a time and replaced with desired functions each time.

What Is Jackknife Resampling?

Jackknife resampling is a statistical method that involves recalculating an estimate derived from subsamples of the provided data samples to confirm its standard error and bias.
Its straightforward resampling techniques seek to offer estimates of an estimate's bias or standard error.
It has advantages when estimating variance and bias related to a given sample data, whereas if the sample becomes large, having high-frequency data samples, then it becomes burdensome.
It is a simpler statistical technique for complex sampling, while bootstrapping is less computationally intensive for large samples.

Jackknife resampling Explained

The jackknife resampling method calculates the estimate's variance and bias in the absence of prohibitive distributional assumptions. It is based on the concept of methodically recomputing a given estimate while excluding one or more observations every time the investigator accesses the sample. As a result, the standard error and bias can be calculated based on a new set of recomputed estimations called jackknife replications.

The jackknife estimator is built for a sample size of n by summating the parameter estimates taken from each subsample of size (n-1). It calculates the sample mean or estimator by sequentially deleting a single observation from the sample. One recomputes the estimator until there are n estimates for a sample size of n. However, its estimated standard error becomes larger than bootstrap resampling.

It has become one of the most used tools by data analysts and statisticians as it is a simple yet powerful resampling method. The technique has been used to estimate a statistic's variance and bias and can easily be applied to complex sampling models. It has also been used in various fields, such as jackknife resampling in R, jackknife resampling in statistics, and jackknife resampling in Python. In the financial world, investors and analysts widely use the jackknife resampling technique in financial modeling and analysis.

Examples

Let us use a few examples to understand the topic.

Example #1

One of the examples of the jackknife model is its application to estimate confidence limits for air quality model evaluations, which is crucial for decision-making in industrial planning and pollution control. Traditional statistical approaches are ineffective due to air quality data's non-Gaussian or non-normal distribution. The jackknife method is applied to air quality model predictions. It provides estimates of confidence limits for air quality model evaluations based on the variability observed across these recalculated performance measures based on different subsamples.

Example #2

Imagine Anna wants to estimate the average daily sales of her retail chain with five stores. One of her stores is much larger than the others. Hence, the probability of skewing the estimate while using each store's weekly sales is high. To address this concern, Anna applied the jackknife method.

The technique involves calculating the average daily sales for all five stores combined. Then, temporarily exclude one store's sales and recalculate the average with the remaining four stores. Repeat this process for each store, excluding one at a time. By comparing all these averages, Anna can determine if removing any individual store significantly changes the average daily sales estimate for the entire retail chain. This helps identify if one store's performance disproportionately influences the overall estimate.

Advantages And Disadvantages

The advantages and disadvantages of jackknife resampling are as follows:

Advantages	Disadvantages
Most helpful in the estimation of variance and bias for a given sample.	It becomes computationally intensive when applied to large samples with data of ultra-high frequency.
Has easier application on complex sampling schemes.	Tends to be more conservative, as it may produce slightly larger estimated standard errors.
On repeated resampling, it gives the same results.	It is a less commonly used method of resampling compared to bootstrapping in practical applications and research.
Help assess variation present in an estimator.	More suitable for small original samples of data.
Aids in reducing bias from a sample.	It may not perform as efficiently as bootstrapping.

Jackknife Resampling vs Bootstrapping

The differences between jackknife resampling and bootstrapping are as follows:

Jackknife resampling	Bootstrapping
Results in the same answer repeatedly when applied to the same data.	This method may yield different results each time.
This method does not involve random sampling of data.	Bootstrapping comprises selecting random samples from the sample observations.
Useful for multistage sampling having varying sampling weights.	This method may not be as efficient for multistage sampling with varying sampling weights.
This method comprises sampling without replacement.	Comprises sampling with replacement.
Larger estimated standard error.	Lower estimated standard error.
Easily applicable when other estimators are difficult to apply or when other methods are unavailable.	This is not used in every situation.

Frequently Asked Questions (FAQs)

1. What is the principle of jackknife resampling?

The jackknife resampling method of estimating a statistic's variability is based on consistently removing one or more observations from a dataset and recalculating the statistic for each subset. This creates various subsamples and their parameter estimates. This approach allows for assessing the statistic's variability without making strong distributional assumptions.

2. Why is it called jackknife resampling?

The term "jackknife" originates from the tool known as a "jackknife," which is a folding knife with a blade that folds back into the handle. Just as the knife's blade can be used for various purposes, jackknife resampling involves repeatedly removing portions of the data to analyze variability.

3. What is the importance of jackknife resampling?

It is an important statistical technique since it allows estimating a statistic's variability without relying on distributional assumptions, making it applicable in various scenarios. It can also be used to analyze the stability and reliability of estimates by systematically assessing the impact of individual observations on the overall statistics.