Resampling Methods in Action: How Bootstrap and Jackknife Improve our Estimates

Author(s): Abinaya Subramaniam

Originally published on Towards AI.

Imagine trying to understand a population based on a small sample. We calculate a statistic, maybe the mean test score of students, the average income of households, or the correlation between two variables. But how confident are we in that number? How much could it vary if we repeated the study?

Traditionally statisticians use formulas for standard errors, confidence intervals, and bias, often assuming that the data follows a specific distribution, like the normal distribution. But real world data is messy. Sometimes, we don’t know the underlying distribution or it might not follow any standard form.

This is where resampling techniques come in. Resampling methods are a powerful way to understand the variability and reliability of statistics using the data itself, without relying on strict assumptions. Two of the most popular resampling techniques are Bootstrap and Jackknife.

What is Resampling?

Resampling is repeatedly reusing your data (sample) to simulate what might happen if you collected data again. Think of your sample as a mini version of the population. By creating new samples from it, you can mimic the process of taking multiple samples from the real population.

Resampling helps answer questions like,

• How variable is my statistic?
• How biased might my estimate be?
• What are reasonable confidence intervals for my estimate?

The beauty of resampling is that it works even when theoretical formulas are difficult or impossible to apply, making it especially useful in modern statistics and machine learning.

The Bootstrap: Simulating Many Samples

The bootstrap is a flexible resampling method introduced by Bradley Efron in 1979. The idea is simple yet powerful. Create many new datasets from your original sample and see how your statistic behaves.

The term bootstrap comes from the phrase “to pull oneself up by one’s bootstraps,” which means to achieve something without external help. In statistics, the bootstrap method reflects this idea. It allows us to estimate the properties of a population like variability, bias, or confidence intervals using only the sample data at hand, without needing to know the true underlying distribution.

Essentially, the method pulls itself up by resampling from the observed data to mimic what repeated sampling from the real population would look like. This self sufficient, data-driven approach is what inspired Bradley Efron to give the method its memorable name.

How it works

1. Take your original dataset.
2. Randomly select observations with replacement to create a new sample of the same size. With replacement means the same observation can appear multiple times in a new sample.
3. Compute the statistic of interest (mean, median, correlation, etc.) for this resample.
4. Repeat steps 2–3 hundreds or thousands of times to generate a distribution of your statistic, called bootstrap replicates.
5. Use the variation in the bootstrap replicates to estimate standard errors, bias, and confidence intervals

By treating your sample as a pseudo population, each resample is like a mini experiment. The variation across the resamples mimics the natural variation you would see if you could repeatedly sample from the true population.

Example

Suppose you have LSAT and GPA scores from 15 law schools. You calculate the correlation between LSAT and GPA as 0.776. How confident are you in this number?

Using the bootstrap, we can,

• Resample the 15 schools many times with replacement.
• Compute the correlation for each resample.
• Look at how the correlation varies across resamples.

The spread of the correlations gives a bootstrap estimate of the standard error. We can also use these replicates to construct confidence intervals, such as the 95% interval, which tells you the range in which the true correlation likely falls.

Bootstrap and Its Connection to Machine Learning

In machine learning, understanding the uncertainty and stability of models is crucial, and the bootstrap provides a natural framework for this. Many ML algorithms, particularly ensemble methods like Random Forests and Bagging, directly rely on the bootstrap idea.

For example, in bagging (Bootstrap Aggregating), multiple decision trees are trained on different bootstrap samples of the training data. Each tree sees a slightly different version of the dataset, and their predictions are averaged (for regression) or voted on (for classification). This resampling reduces variance, making the model more robust and less prone to overfitting.

The Jackknife: Leave-One-Out Insights

The jackknife predates the bootstrap and was proposed by Quenouille and Tukey decades ago. It’s a simpler resampling method and works as a leave-one-out technique.

The jackknife gets its name from the concept of a jackknife tool, which is small, versatile, and useful for many tasks. Similarly, the jackknife method is a simple but powerful resampling technique that can be applied to a variety of statistics to estimate bias and standard error.

The term also reflects the method’s leave-one-out approach, where each observation is temporarily removed from the dataset, much like how a jackknife can be opened and used one blade at a time. Its elegance and utility in handling small datasets and smooth statistics made the name a fitting metaphor for this early resampling technique.

How it works

1. Take your original dataset of n observations.
2. Create n new datasets, each leaving out exactly one observation.
3. Compute the statistic of interest for each of these leave-one-out samples. These are called jackknife replicates.
4. Analyze the variability of the replicates to estimate bias and standard error.

Why it works

The jackknife examines how sensitive a statistic is to individual observations. If leaving out one observation changes the estimate significantly, the statistic is highly sensitive and may have a higher standard error. If it changes very little, the statistic is stable.

Example

Suppose you have 5 test scores: 160, 165, 170, 175, 180. The mean is 170.

• Remove the first score: mean of remaining 4 = 167.5
• Remove the second score: mean = 166.25
• Continue for all scores.

The variation in these leave one out means gives the jackknife estimate of standard error. We can also estimate bias by comparing the average of these replicates to the original mean.

Why the Jackknife Can Fail

The jackknife relies on the principle of leave-one-out resampling, which works well for smooth statistics, those that change gradually when individual observations are removed. For example, the sample mean or variance adjusts predictably when one data point is omitted, allowing the jackknife to accurately estimate standard errors and bias.

However, not all statistics are smooth. The median, minimum, maximum, or other extreme quantiles are highly insensitive to the removal of a single observation. In these cases, leaving out one value may not change the statistic at all, or it may change it in a very irregular way.

As a result, the variability of the jackknife replicates does not reflect the true variability of the statistic leading to misleadingly low estimates of standard error.

This limitation becomes particularly apparent in small datasets or when the data contain outliers. For example, if you calculate the jackknife standard error for the median of a small sample of numbers most of the leave-one-out medians might be identical, producing a standard error close to zero, even though the true sampling variability of the median could be substantial.

To address this, statisticians may use the bootstrap, which resamples with replacement and can handle non-smooth statistics more reliably or the delete-d jackknife, which removes multiple observations at a time to better capture variability. Understanding when the jackknife is likely to fail is crucial to avoid overconfidence in results based on inappropriate resampling assumptions.

When to Use Bootstrap vs. Jackknife

Choosing between the bootstrap and the jackknife depends on the type of statistic you are analyzing and the goals of your analysis. The jackknife works best for smooth statistics such as the mean, variance, or regression coefficients, where small changes in the data produce small changes in the statistic.

It is also ideal for small to moderate datasets when we want a quick and computationally light estimate of bias or standard error. The jackknife does not generate a full distribution of the statistic, so it is most useful when all you need is a simple measure of variability or bias.

On the other hand, the bootstrap is a more flexible and powerful method that can handle almost any statistic, including non smooth ones like the median, quantiles, or complex estimators. It is particularly useful when you want to estimate standard errors, bias, or confidence intervals using the full distribution of replicates.

The bootstrap is well suited for larger datasets where computational cost is less of an issue and is also widely used in machine learning, powering ensemble methods like bagging and random forests. In short, while the jackknife is fast and straightforward for simpler problems, the bootstrap provides versatility and robustness for analyzing complex statistics and understanding the variability of your estimates in depth.

Join thousands of data leaders on the AI newsletter. Join over 80,000 subscribers and keep up to date with the latest developments in AI. From research to projects and ideas. If you are building an AI startup, an AI-related product, or a service, we invite you to consider becoming a sponsor.

Published via Towards AI