Bootstrapping (statistics)

Illustration bootstrap

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Poisson approximation to Binomial

|MedianHists|thumb|left]]|thumb|left]] Bootstrapping (statistics) is a resampling method used in statistics to estimate the distribution of a sample statistic. It involves repeatedly drawing samples, with replacement, from an observed dataset and calculating the statistic of interest for each sample. This method allows for the estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping is a powerful tool because it does not rely on the assumption of normality and can be applied in situations where the theoretical distribution of the statistic is unknown or difficult to derive.

Overview

Bootstrapping was introduced by Bradley Efron in 1979 and has since become a fundamental technique in statistical inference. The basic idea is to create a "bootstrap" sample by randomly selecting observations from the original dataset with replacement. This process is repeated a large number of times (typically thousands or more), and for each bootstrap sample, the desired statistic is computed. The collection of these statistics forms an empirical distribution, which can then be used to estimate properties such as the mean, variance, confidence intervals, and hypothesis testing.

Procedure

The general procedure for bootstrapping involves several steps:

1. From the original dataset of size n, draw a sample of size n with replacement. This sample is known as a bootstrap sample.
2. Calculate the statistic of interest for the bootstrap sample.
3. Repeat steps 1 and 2 a large number of times (e.g., 1000 or 10000 times) to create a distribution of the bootstrap statistics.
4. Use the distribution of bootstrap statistics to estimate the desired characteristics of the sampling distribution of the statistic (e.g., its variance, bias, confidence intervals).

Types of Bootstrapping

There are several types of bootstrapping methods, including but not limited to:

• Non-parametric bootstrapping: The most straightforward form, which does not assume any specific parametric form for the data distribution.
• Parametric bootstrapping: Assumes that the data follow a certain distribution and samples are drawn from that distribution instead of the original dataset.
• Block bootstrapping: Used for data that are correlated over time, such as time series data, where blocks of data are resampled instead of individual observations.

Applications

Bootstrapping is used in various statistical applications, including:

• Estimating the distribution of a sample mean or median
• Constructing confidence intervals for a population parameter
• Hypothesis testing
• Model selection and validation in machine learning

Advantages and Limitations

Advantages:

• Does not require the assumption of normality or other specific distributional assumptions.
• Can be applied to complex estimators or those with no closed-form distribution.
• Useful in situations with small sample sizes.

Limitations:

• Bootstrap methods can be computationally intensive, especially with large datasets and a high number of resampling iterations.
• Not always appropriate for data with strong dependencies, such as time series, without modifications (e.g., block bootstrapping).
• The accuracy of bootstrap confidence intervals can depend on the choice of the method (e.g., percentile method, BCa method).

Conclusion

Bootstrapping is a versatile and powerful statistical tool that has broad applications in statistical inference. Its ability to estimate the distribution of a statistic without relying on strict assumptions about the population from which the sample is drawn makes it invaluable in practical statistical analysis and research.