Bootstrapping is any test or metric that uses random sampling with replacement (e.g. mimicking the sampling process), and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

Bootstrapping estimates the properties of an estimand (such as its variance) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution function of the observed data. In the case where a set of observations can be assumed to be from an independent and identically distributed population, this can be implemented by constructing a number of resamples with replacement, of the observed data set (and of equal size to the observed data set).

It may also be used for constructing hypothesis tests. It is often used as an alternative to statistical inference based on the assumption of a parametric model when that assumption is in doubt, or where parametric inference is impossible or requires complicated formulas for the calculation of standard errors.

## History

The bootstrap was published by Bradley Efron in "Bootstrap methods: another look at the jackknife" (1979), inspired by earlier work on the jackknife. Improved estimates of the variance were developed later. A Bayesian extension was developed in 1981. The bias-corrected and accelerated (BCa) bootstrap was developed by Efron in 1987, and the ABC procedure in 1992.

## Approach

The basic idea of bootstrapping is that inference about a population from sample data (sample → population) can be modeled by resampling the sample data and performing inference about a sample from resampled data (resampled → sample). As the population is unknown, the true error in a sample statistic against its population value is unknown. In bootstrap-resamples, the 'population' is in fact the sample, and this is known; hence the quality of inference of the 'true' sample from resampled data (resampled → sample) is measurable.

More formally, the bootstrap works by treating inference of the true probability distribution J, given the original data, as being analogous to an inference of the empirical distribution Ĵ, given the resampled data. The accuracy of inferences regarding Ĵ using the resampled data can be assessed because we know Ĵ. If Ĵ is a reasonable approximation to J, then the quality of inference on J can in turn be inferred.

As an example, assume we are interested in the average (or mean) height of people worldwide. We cannot measure all the people in the global population, so instead, we sample only a tiny part of it, and measure that. Assume the sample is of size N; that is, we measure the heights of N individuals. From that single sample, only one estimate of the mean can be obtained. In order to reason about the population, we need some sense of the variability of the mean that we have computed. The simplest bootstrap method involves taking the original data set of heights, and, using a computer, sampling from it to form a new sample (called a 'resample' or bootstrap sample) that is also of size N. The bootstrap sample is taken from the original by using sampling with replacement (e.g. we might 'resample' 5 times from [1,2,3,4,5] and get [2,5,4,4,1]), so, assuming N is sufficiently large, for all practical purposes there is virtually zero probability that it will be identical to the original "real" sample. This process is repeated a large number of times (typically 1,000 or 10,000 times), and for each of these bootstrap samples, we compute its mean (each of these is called a "bootstrap estimate"). We now can create a histogram of bootstrap means. This histogram provides an estimate of the shape of the distribution of the sample mean from which we can answer questions about how much the mean varies across samples. (The method here, described for the mean, can be applied to almost any other statistic or estimator.)

## Discussion

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A great advantage of bootstrap is its simplicity. It is a straightforward way to derive estimates of standard errors and confidence intervals for complex estimators of the distribution, such as percentile points, proportions, odds ratio, and correlation coefficients. However, despite its simplicity, bootstrapping can be applied to complex sampling designs (e.g. for population divided into s strata with ns observations per strata, bootstrapping can be applied for each strata). Bootstrap is also an appropriate way to control and check the stability of the results. Although for most problems it is impossible to know the true confidence interval, bootstrap is asymptotically more accurate than the standard intervals obtained using sample variance and assumptions of normality. Bootstrapping is also a convenient method that avoids the cost of repeating the experiment to get other groups of sample data.

Bootstrapping depends heavily on the estimator used and, though simple, naive use of bootstrapping will not always yield asymptotically valid results and can lead to inconsistency. Although bootstrapping is (under some conditions) asymptotically consistent, it does not provide general finite-sample guarantees. The result may depend on the representative sample. The apparent simplicity may conceal the fact that important assumptions are being made when undertaking the bootstrap analysis (e.g. independence of samples or large enough of a sample size) where these would be more formally stated in other approaches. Also, bootstrapping can be time-consuming and there are not many available software for bootstrapping as it is difficult to automate using traditional statistical computer packages.

### Recommendations

Scholars have recommended more bootstrap samples as available computing power has increased. If the results may have substantial real-world consequences, then one should use as many samples as is reasonable, given available computing power and time. Increasing the number of samples cannot increase the amount of information in the original data; it can only reduce the effects of random sampling errors which can arise from a bootstrap procedure itself. Moreover, there is evidence that numbers of samples greater than 100 lead to negligible improvements in the estimation of standard errors. In fact, according to the original developer of the bootstrapping method, even setting the number of samples at 50 is likely to lead to fairly good standard error estimates.

Adèr et al. recommend the bootstrap procedure for the following situations:

• When the theoretical distribution of a statistic of interest is complicated or unknown. Since the bootstrapping procedure is distribution-independent it provides an indirect method to assess the properties of the distribution underlying the sample and the parameters of interest that are derived from this distribution.
• When the sample size is insufficient for straightforward statistical inference. If the underlying distribution is well-known, bootstrapping provides a way to account for the distortions caused by the specific sample that may not be fully representative of the population.
• When power calculations have to be performed, and a small pilot sample is available. Most power and sample size calculations are heavily dependent on the standard deviation of the statistic of interest. If the estimate used is incorrect, the required sample size will also be wrong. One method to get an impression of the variation of the statistic is to use a small pilot sample and perform bootstrapping on it to get impression of the variance.

However, Athreya has shown that if one performs a naive bootstrap on the sample mean when the underlying population lacks a finite variance (for example, a power law distribution), then the bootstrap distribution will not converge to the same limit as the sample mean. As a result, confidence intervals on the basis of a Monte Carlo simulation of the bootstrap could be misleading. Athreya states that "Unless one is reasonably sure that the underlying distribution is not heavy tailed, one should hesitate to use the naive bootstrap".

## Types of bootstrap scheme

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In univariate problems, it is usually acceptable to resample the individual observations with replacement ("case resampling" below) unlike subsampling, in which resampling is without replacement and is valid under much weaker conditions compared to the bootstrap. In small samples, a parametric bootstrap approach might be preferred. For other problems, a smooth bootstrap will likely be preferred.

For regression problems, various other alternatives are available.

### Case resampling

The bootstrap is generally useful for estimating the distribution of a statistic (e.g. mean, variance) without using normality assumptions (as required, e.g., for a z-statistic or a t-statistic). In particular, the bootstrap is useful when there is no analytical form or an asymptotic theory (e.g., an applicable central limit theorem) to help estimate the distribution of the statistics of interest. This is because bootstrap methods can apply to most random quantities, e.g., the ratio of variance and mean. There are at least two ways of performing case resampling.

1. The Monte Carlo algorithm for case resampling is quite simple. First, we resample the data with replacement, and the size of the resample must be equal to the size of the original data set. Then the statistic of interest is computed from the resample from the first step. We repeat this routine many times to get a more precise estimate of the Bootstrap distribution of the statistic.
2. The 'exact' version for case resampling is similar, but we exhaustively enumerate every possible resample of the data set. This can be computationally expensive as there are a total of ${\binom {2n-1}{n))={\frac {(2n-1)!}{n!(n-1)!))$ different resamples, where n is the size of the data set. Thus for n = 5, 10, 20, 30 there are 126, 92378, 6.89 × 1010 and 5.91 × 1016 different resamples respectively.

#### Estimating the distribution of sample mean

Consider a coin-flipping experiment. We flip the coin and record whether it lands heads or tails. Let X = x1, x2, …, x10 be 10 observations from the experiment. xi = 1 if the i th flip lands heads, and 0 otherwise. By invoking the assumption that the average of the coin flips is normally distributed, we can use the t-statistic to estimate the distribution of the sample mean,

${\bar {x))={\frac {1}{10))(x_{1}+x_{2}+\cdots +x_{10}).$ Such a normality assumption can be justified either as an approximation of the distribution of each individual coin flip or as an approximation of the distribution of the average of a large number of coin flips. The former is a poor approximation because the true distribution of the coin flips is Bernoulli instead of normal. The latter is a valid approximation in infinitely large samples due to the central limit theorem.

However, if we are not ready to make such a justification, then we can use the bootstrap instead. Using case resampling, we can derive the distribution of ${\bar {x))$ . We first resample the data to obtain a bootstrap resample. An example of the first resample might look like this X1* = x2, x1, x10, x10, x3, x4, x6, x7, x1, x9. There are some duplicates since a bootstrap resample comes from sampling with replacement from the data. Also the number of data points in a bootstrap resample is equal to the number of data points in our original observations. Then we compute the mean of this resample and obtain the first bootstrap mean: μ1*. We repeat this process to obtain the second resample X2* and compute the second bootstrap mean μ2*. If we repeat this 100 times, then we have μ1*, μ2*, ..., μ100*. This represents an empirical bootstrap distribution of sample mean. From this empirical distribution, one can derive a bootstrap confidence interval for the purpose of hypothesis testing.

#### Regression

In regression problems, case resampling refers to the simple scheme of resampling individual cases – often rows of a data set. For regression problems, as long as the data set is fairly large, this simple scheme is often acceptable. However, the method is open to criticism[citation needed].

In regression problems, the explanatory variables are often fixed, or at least observed with more control than the response variable. Also, the range of the explanatory variables defines the information available from them. Therefore, to resample cases means that each bootstrap sample will lose some information. As such, alternative bootstrap procedures should be considered.

### Bayesian bootstrap

Bootstrapping can be interpreted in a Bayesian framework using a scheme that creates new data sets through reweighting the initial data. Given a set of $N$ data points, the weighting assigned to data point $i$ in a new data set ${\mathcal {D))^{J)$ is $w_{i}^{J}=x_{i}^{J}-x_{i-1}^{J)$ , where $\mathbf {x} ^{J)$ is a low-to-high ordered list of $N-1$ uniformly distributed random numbers on $[0,1]$ , preceded by 0 and succeeded by 1. The distributions of a parameter inferred from considering many such data sets ${\mathcal {D))^{J)$ are then interpretable as posterior distributions on that parameter.

### Smooth bootstrap

Under this scheme, a small amount of (usually normally distributed) zero-centered random noise is added onto each resampled observation. This is equivalent to sampling from a kernel density estimate of the data. Assume K to be a symmetric kernel density function with unit variance. The standard kernel estimator ${\hat {f\,))_{h}(x)$ of $f(x)$ is

${\hat {f\,))_{h}(x)={1 \over nh}\sum _{i=1}^{n}K\left({x-X_{i} \over h}\right),$ where $h$ is the smoothing parameter. And the corresponding distribution function estimator ${\hat {F\,))_{h}(x)$ is

${\hat {F\,))_{h}(x)=\int _{-\infty }^{x}{\hat {f))_{h}(t)\,dt.$ ### Parametric bootstrap

Based on the assumption that the original data set is a realization of a random sample from a distribution of a specific parametric type, in this case a parametric model is fitted by parameter θ, often by maximum likelihood, and samples of random numbers are drawn from this fitted model. Usually the sample drawn has the same sample size as the original data. Then the estimate of original function F can be written as ${\hat {F))=F_{\hat {\theta ))$ . This sampling process is repeated many times as for other bootstrap methods. Considering the centered sample mean in this case, the random sample original distribution function $F_{\theta )$ is replaced by a bootstrap random sample with function $F_{\hat {\theta ))$ , and the probability distribution of ${\bar {X_{n))}-\mu _{\theta )$ is approximated by that of ${\bar {X))_{n}^{*}-\mu ^{*)$ , where $\mu ^{*}=\mu _{\hat {\theta ))$ , which is the expectation corresponding to $F_{\hat {\theta ))$ . The use of a parametric model at the sampling stage of the bootstrap methodology leads to procedures which are different from those obtained by applying basic statistical theory to inference for the same model.

### Resampling residuals

Another approach to bootstrapping in regression problems is to resample residuals. The method proceeds as follows.

1. Fit the model and retain the fitted values ${\widehat {y\,))_{i)$ and the residuals ${\widehat {\varepsilon \,))_{i}=y_{i}-{\widehat {y\,))_{i},(i=1,\dots ,n)$ .
2. For each pair, (xi, yi), in which xi is the (possibly multivariate) explanatory variable, add a randomly resampled residual, ${\widehat {\varepsilon \,))_{j)$ , to the fitted value ${\widehat {y\,))_{i)$ . In other words, create synthetic response variables $y_{i}^{*}={\widehat {y\,))_{i}+{\widehat {\varepsilon \,))_{j)$ where j is selected randomly from the list (1, ..., n) for every i.
3. Refit the model using the fictitious response variables $y_{i}^{*)$ , and retain the quantities of interest (often the parameters, ${\widehat {\mu ))_{i}^{*)$ , estimated from the synthetic $y_{i}^{*)$ ).
4. Repeat steps 2 and 3 a large number of times.

This scheme has the advantage that it retains the information in the explanatory variables. However, a question arises as to which residuals to resample. Raw residuals are one option; another is studentized residuals (in linear regression). Although there are arguments in favor of using studentized residuals; in practice, it often makes little difference, and it is easy to compare the results of both schemes.

### Gaussian process regression bootstrap

When data are temporally correlated, straightforward bootstrapping destroys the inherent correlations. This method uses Gaussian process regression (GPR) to fit a probabilistic model from which replicates may then be drawn. GPR is a Bayesian non-linear regression method. A Gaussian process (GP) is a collection of random variables, any finite number of which have a joint Gaussian (normal) distribution. A GP is defined by a mean function and a covariance function, which specify the mean vectors and covariance matrices for each finite collection of the random variables.

Regression model:

$y(x)=f(x)+\varepsilon ,\ \ \varepsilon \sim {\mathcal {N))(0,\sigma ^{2}),$ $\varepsilon$ is a noise term.

Gaussian process prior:

For any finite collection of variables, x1, ..., xn, the function outputs $f(x_{1}),\ldots ,f(x_{n})$ are jointly distributed according to a multivariate Gaussian with mean $m=[m(x_{1}),\ldots ,m(x_{n})]^{\intercal )$ and covariance matrix $(K)_{ij}=k(x_{i},x_{j}).$ Assume $f(x)\sim {\mathcal {GP))(m,k).$ Then $y(x)\sim {\mathcal {GP))(m,l)$ ,

where $l(x_{i},x_{j})=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j})$ , and $\delta (x_{i},x_{j})$ is the standard Kronecker delta function.

Gaussian process posterior:

According to GP prior, we can get

$[y(x_{1}),\ldots ,y(x_{r})]\sim {\mathcal {N))(m_{0},K_{0})$ ,

where $m_{0}=[m(x_{1}),\ldots ,m(x_{r})]^{\intercal )$ and $(K_{0})_{ij}=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j}).$ Let x1*,...,xs* be another finite collection of variables, it's obvious that

$[y(x_{1}),\ldots ,y(x_{r}),f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\sim {\mathcal {N))({\binom {m_{0)){m_{*))}{\begin{pmatrix}K_{0}&K_{*}\\K_{*}^{\intercal }&K_{**}\end{pmatrix)))$ ,

where $m_{*}=[m(x_{1}^{*}),\ldots ,m(x_{s}^{*})]^{\intercal )$ , $(K_{**})_{ij}=k(x_{i}^{*},x_{j}^{*})$ , $(K_{*})_{ij}=k(x_{i},x_{j}^{*}).$ According to the equations above, the outputs y are also jointly distributed according to a multivariate Gaussian. Thus,

$[f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\mid ([y(x)]^{\intercal }=y)\sim {\mathcal {N))(m_{\text{post)),K_{\text{post))),$ where $y=[y_{1},...,y_{r}]^{\intercal )$ , $m_{\text{post))=m_{*}+K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}(y-m_{0})$ , $K_{\text{post))=K_{**}-K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}K_{*)$ , and $I_{r)$ is $r\times r$ identity matrix.

### Wild bootstrap

The wild bootstrap, proposed originally by Wu (1986), is suited when the model exhibits heteroskedasticity. The idea is, as the residual bootstrap, to leave the regressors at their sample value, but to resample the response variable based on the residuals values. That is, for each replicate, one computes a new $y$ based on

$y_{i}^{*}={\widehat {y\,))_{i}+{\widehat {\varepsilon \,))_{i}v_{i)$ so the residuals are randomly multiplied by a random variable $v_{i)$ with mean 0 and variance 1. For most distributions of $v_{i)$ (but not Mammen's), this method assumes that the 'true' residual distribution is symmetric and can offer advantages over simple residual sampling for smaller sample sizes. Different forms are used for the random variable $v_{i)$ , such as

• A distribution suggested by Mammen (1993).

## Example applications

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### Smoothed bootstrap

In 1878, Simon Newcomb took observations on the speed of light. The data set contains two outliers, which greatly influence the sample mean. (The sample mean need not be a consistent estimator for any population mean, because no mean needs to exist for a heavy-tailed distribution.) A well-defined and robust statistic for the central tendency is the sample median, which is consistent and median-unbiased for the population median.

The bootstrap distribution for Newcomb's data appears below. We can reduce the discreteness of the bootstrap distribution by adding a small amount of random noise to each bootstrap sample. A conventional choice is to add noise with a standard deviation of $\sigma /{\sqrt {n))$ for a sample size n; this noise is often drawn from a Student-t distribution with n-1 degrees of freedom. This results in an approximately-unbiased estimator for the variance of the sample mean. This means that samples taken from the bootstrap distribution will have a variance which is, on average, equal to the variance of the total population.

Histograms of the bootstrap distribution and the smooth bootstrap distribution appear below. The bootstrap distribution of the sample-median has only a small number of values. The smoothed bootstrap distribution has a richer support. However, note that whether the smoothed or standard bootstrap procedure is favorable is case-by-case and is shown to depend on both the underlying distribution function and on the quantity being estimated. In this example, the bootstrapped 95% (percentile) confidence-interval for the population median is (26, 28.5), which is close to the interval for (25.98, 28.46) for the smoothed bootstrap.

## Relation to other approaches to inference

### Relationship to other resampling methods

The bootstrap is distinguished from:

• the jackknife procedure, used to estimate biases of sample statistics and to estimate variances, and
• cross-validation, in which the parameters (e.g., regression weights, factor loadings) that are estimated in one subsample are applied to another subsample.

For more details see resampling.

Bootstrap aggregating (bagging) is a meta-algorithm based on averaging model predictions obtained from models trained on multiple bootstrap samples.

### U-statistics

 Main article: U-statistic

In situations where an obvious statistic can be devised to measure a required characteristic using only a small number, r, of data items, a corresponding statistic based on the entire sample can be formulated. Given an r-sample statistic, one can create an n-sample statistic by something similar to bootstrapping (taking the average of the statistic over all subsamples of size r). This procedure is known to have certain good properties and the result is a U-statistic. The sample mean and sample variance are of this form, for r = 1 and r = 2.

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