Bootstrapping (statistics)

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. 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.[16] Bootstrapping is also a convenient method that avoids the cost of repeating the experiment to get other groups of sample data.

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) where these would be more formally stated in other approaches. Also, bootstrapping can be time-consuming.

The number of bootstrap samples recommended in literature has increased 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.[17] 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.[18]

Adèr et al. recommend the bootstrap procedure for the following situations:[19]

• 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[20] 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".

Bootstrap is generally useful for estimating the distribution of a statistic (e.g. mean, variance) without using normal theory (e.g. z-statistic, t-statistic). Bootstrap comes in handy when there is no analytical form or normal theory to help estimate the distribution of the statistics of interest, since 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 ${\displaystyle {\binom {2n-1}{n}}={\frac {(2n-1)!}{n!(n-1)!}}}$ 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.[22]

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. x_i = 1 if the i th flip lands heads, and 0 otherwise. From normal theory, we can use t-statistic to estimate the distribution of the sample mean,

${\displaystyle {\bar {x}}={\frac {1}{10}}(x_{1}+x_{2}+\cdots +x_{10}).}$

Instead, we use bootstrap, specifically case resampling, to derive the distribution of ${\displaystyle {\bar {x}}}$. 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.

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.

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.

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 ${\displaystyle N}$ data points, the weighting assigned to data point ${\displaystyle i}$ in a new data set ${\displaystyle {\mathcal {D}}^{J}}$ is ${\displaystyle w_{i}^{J}=x_{i}^{J}-x_{i-1}^{J}}$, where ${\displaystyle \mathbf {x} ^{J}}$ is a low-to-high ordered list of ${\displaystyle N-1}$ uniformly distributed random numbers on ${\displaystyle [0,1]}$, preceded by 0 and succeeded by 1. The distributions of a parameter inferred from considering many such data sets ${\displaystyle {\mathcal {D}}^{J}}$ are then interpretable as posterior distributions on that parameter.[23]

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 ${\displaystyle {\hat {f\,}}_{h}(x)}$ of ${\displaystyle f(x)}$ is

${\displaystyle {\hat {f\,}}_{h}(x)={1 \over nh}\sum _{i=1}^{n}K\left({x-X_{i} \over h}\right),}$ [24]

where ${\displaystyle h}$ is the smoothing parameter. And the corresponding distribution function estimator ${\displaystyle {\hat {F\,}}_{h}(x)}$ is

${\displaystyle {\hat {F\,}}_{h}(x)=\int _{-\infty }^{x}{\hat {f}}_{h}(t)\,dt.}$ [24]

Based on the assumption that original data set is 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 ${\displaystyle {\hat {F}}=F_{\hat {\theta }}}$. 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 ${\displaystyle F_{\theta }}$ is replaced by a bootstrap random sample with function ${\displaystyle F_{\hat {\theta }}}$, and the probability distribution of ${\displaystyle {\bar {X_{n}}}-\mu _{\theta }}$ is approximated by that of ${\displaystyle {\bar {X}}_{n}^{*}-\mu ^{*}}$, where ${\displaystyle \mu ^{*}=\mu _{\hat {\theta }}}$, which is the expectation corresponding to ${\displaystyle F_{\hat {\theta }}}$.[25] 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.

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 ${\displaystyle {\widehat {y\,}}_{i}}$ and the residuals ${\displaystyle {\widehat {\varepsilon \,}}_{i}=y_{i}-{\widehat {y\,}}_{i},(i=1,\dots ,n)}$.
2. For each pair, (x_i, y_i), in which x_i is the (possibly multivariate) explanatory variable, add a randomly resampled residual, ${\displaystyle {\widehat {\varepsilon \,}}_{j}}$, to the fitted value ${\displaystyle {\widehat {y\,}}_{i}}$. In other words, create synthetic response variables ${\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{j}}$ where j is selected randomly from the list (1, ..., n) for every i.
3. Refit the model using the fictitious response variables ${\displaystyle y_{i}^{*}}$, and retain the quantities of interest (often the parameters, ${\displaystyle {\widehat {\mu }}_{i}^{*}}$, estimated from the synthetic ${\displaystyle y_{i}^{*}}$).
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 favour of using studentized residuals; in practice, it often makes little difference, and it is easy to compare the results of both schemes.

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, and 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. [26]

Regression model:

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

Gaussian process prior:

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

Assume ${\displaystyle f(x)\sim {\mathcal {GP}}(m,k).}$ Then ${\displaystyle y(x)\sim {\mathcal {GP}}(m,l)}$,

where ${\displaystyle l(x_{i},x_{j})=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j})}$, and ${\displaystyle \delta (x_{i},x_{j})}$ is the standard Kronecker delta function.[26]

Gaussian process posterior:

According to GP prior, we can get

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

where ${\displaystyle m_{0}=[m(x_{1}),\ldots ,m(x_{r})]^{\intercal }}$ and ${\displaystyle (K_{0})_{ij}=k(x_{i},x_{j})+\sigma ^{2}\delta (x_{i},x_{j}).}$

Let x₁^*,...,x_s^* be another finite collection of variables, it's obvious that

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

where ${\displaystyle m_{*}=[m(x_{1}^{*}),\ldots ,m(x_{s}^{*})]^{\intercal }}$, ${\displaystyle (K_{**})_{ij}=k(x_{i}^{*},x_{j}^{*})}$, ${\displaystyle (K_{*})_{ij}=k(x_{i},x_{j}^{*}).}$

According to the equations above, the outputs y are also jointly distributed according to a multivariate Gaussian. Thus,

${\displaystyle [f(x_{1}^{*}),\ldots ,f(x_{s}^{*})]^{\intercal }\mid ([y(x)]^{\intercal }=y)\sim {\mathcal {N}}(m_{\text{post}},K_{\text{post}}),}$

where ${\displaystyle y=[y_{1},...,y_{r}]^{\intercal }}$, ${\displaystyle m_{\text{post}}=m_{*}+K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}(y-m_{0})}$, ${\displaystyle K_{\text{post}}=K_{**}-K_{*}^{\intercal }(K_{O}+\sigma ^{2}I_{r})^{-1}K_{*}}$, and ${\displaystyle I_{r}}$ is ${\displaystyle r\times r}$ identity matrix.[26]

The wild bootstrap, proposed originally by Wu (1986),[27] is suited when the model exhibits heteroskedasticity. The idea is, like 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 ${\displaystyle y}$ based on

${\displaystyle y_{i}^{*}={\widehat {y\,}}_{i}+{\widehat {\varepsilon \,}}_{i}v_{i}}$

so the residuals are randomly multiplied by a random variable ${\displaystyle v_{i}}$ with mean 0 and variance 1. For most distributions of ${\displaystyle v_{i}}$ (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 ${\displaystyle v_{i}}$, such as

• The standard normal distribution

• A distribution suggested by Mammen (1993).[28]

${\displaystyle v_{i}={\begin{cases}-({\sqrt {5}}-1)/2&{\text{with probability }}({\sqrt {5}}+1)/(2{\sqrt {5}}),\\({\sqrt {5}}+1)/2&{\text{with probability }}({\sqrt {5}}-1)/(2{\sqrt {5}})\end{cases}}}$

Approximately, Mammen's distribution is:

${\displaystyle v_{i}={\begin{cases}-0.6180\quad {\text{(with a 0 in the units' place)}}&{\text{with probability }}0.7236,\\+1.6180\quad {\text{(with a 1 in the units' place)}}&{\text{with probability }}0.2764.\end{cases}}}$

• Or the simpler distribution, linked to the Rademacher distribution:

${\displaystyle v_{i}={\begin{cases}-1&{\text{with probability }}1/2,\\+1&{\text{with probability }}1/2.\end{cases}}}$

The block bootstrap is used when the data, or the errors in a model, are correlated. In this case, a simple case or residual resampling will fail, as it is not able to replicate the correlation in the data. The block bootstrap tries to replicate the correlation by resampling inside blocks of data. The block bootstrap has been used mainly with data correlated in time (i.e. time series) but can also be used with data correlated in space, or among groups (so-called cluster data).

In the (simple) block bootstrap, the variable of interest is split into non-overlapping blocks.

In the moving block bootstrap, introduced by Künsch (1989),[29] data is split into n − b + 1 overlapping blocks of length b: Observation 1 to b will be block 1, observation 2 to b + 1 will be block 2, etc. Then from these n − b + 1 blocks, n/b blocks will be drawn at random with replacement. Then aligning these n/b blocks in the order they were picked, will give the bootstrap observations.

This bootstrap works with dependent data, however, the bootstrapped observations will not be stationary anymore by construction. But, it was shown that varying randomly the block length can avoid this problem.[30] This method is known as the stationary bootstrap. Other related modifications of the moving block bootstrap are the Markovian bootstrap and a stationary bootstrap method that matches subsequent blocks based on standard deviation matching.

Vinod (2006),[31] presents a method that bootstraps time series data using maximum entropy principles satisfying the Ergodic theorem with mean-preserving and mass-preserving constraints. There is an R package, meboot,[32] that utilizes the method, which has applications in econometrics and computer science.

Cluster data describes data where many observations per unit are observed. This could be observing many firms in many states, or observing students in many classes. In such cases, the correlation structure is simplified, and one does usually make the assumption that data is correlated within a group/cluster, but independent between groups/clusters. The structure of the block bootstrap is easily obtained (where the block just corresponds to the group), and usually only the groups are resampled, while the observations within the groups are left unchanged. Cameron et al. (2008) discusses this for clustered errors in linear regression.[33]

Graph showing the convergence of the Binomial Distribution to the Poisson as the Binomial parameters are n*p=1 and n grows

The ordinary bootstrap requires the random selection of n elements from a list, which is equivalent to drawing from a multinomial distribution. This may require a large number of passes over the data and is challenging to run these computations in parallel. For large values of n, the Poisson bootstrap is an efficient method of generating bootstrapped data sets.[34] When generating a single bootstrap sample, instead of randomly drawing from the sample data with replacement, each data point is assigned a random weight distributed according to the Poisson distribution with ${\displaystyle \lambda =1}$. For large sample data, this will approximate random sampling with replacement. This is due to the following approximation:

${\displaystyle \lim _{n\to \infty }\operatorname {Binomial} (n,1/n)=\operatorname {Poisson} (1)}$

This method also lends itself well to streaming data and growing data sets, since the total number of samples does not need to be known in advance of beginning to take bootstrap samples.

For massive data sets, it is often computationally prohibitive to hold all the sample data in memory and resample from the sample data. The Bag of Little Bootstraps (BLB)[35] provides a method of pre-aggregating data before bootstrapping to reduce computational constraints. This works by partitioning the data set into ${\displaystyle b}$ equal sized buckets and aggregating the data within each bucket. This pre-aggregated data set becomes the new sample data over which to draw samples with replacement. This method is similar to the Block Bootstrap, but the motivations and definitions of the blocks are very different. Under certain assumptions, the sample distribution should approximate the full bootstrapped scenario. One constraint is the number of buckets ${\displaystyle b=n^{\gamma }}$where ${\displaystyle \gamma \in [0.5,1]}$ and the authors recommend usage of ${\displaystyle b=n^{0.7}}$ as a general solution.

• Bias: The bootstrap distribution and the sample may disagree systematically, in which case bias may occur.

  If the bootstrap distribution of an estimator is symmetric, then percentile confidence-interval are often used; such intervals are appropriate especially for median-unbiased estimators of minimum risk (with respect to an absolute loss function). Bias in the bootstrap distribution will lead to bias in the confidence-interval.

  Otherwise, if the bootstrap distribution is non-symmetric, then percentile confidence-intervals are often inappropriate.

There are several methods for constructing confidence intervals from the bootstrap distribution of a real parameter:

• Basic bootstrap,[36] also known as the Reverse Percentile Interval.[37] The basic bootstrap is a simple scheme to construct the confidence interval: one simply takes the empirical quantiles from the bootstrap distribution of the parameter (see Davison and Hinkley 1997, equ. 5.6 p. 194):

${\displaystyle (2{\widehat {\theta \,}}-\theta _{(1-\alpha /2)}^{*},2{\widehat {\theta \,}}-\theta _{(\alpha /2)}^{*})}$ where ${\displaystyle \theta _{(1-\alpha /2)}^{*}}$ denotes the ${\displaystyle 1-\alpha /2}$ percentile of the bootstrapped coefficients ${\displaystyle \theta ^{*}}$.

• Percentile bootstrap. The percentile bootstrap proceeds in a similar way to the basic bootstrap, using percentiles of the bootstrap distribution, but with a different formula (note the inversion of the left and right quantiles!):

${\displaystyle (\theta _{(\alpha /2)}^{*},\theta _{(1-\alpha /2)}^{*})}$ where ${\displaystyle \theta _{(1-\alpha /2)}^{*}}$ denotes the ${\displaystyle 1-\alpha /2}$ percentile of the bootstrapped coefficients ${\displaystyle \theta ^{*}}$.

See Davison and Hinkley (1997, equ. 5.18 p. 203) and Efron and Tibshirani (1993, equ 13.5 p. 171).

This method can be applied to any statistic. It will work well in cases where the bootstrap distribution is symmetrical and centered on the observed statistic[38] and where the sample statistic is median-unbiased and has maximum concentration (or minimum risk with respect to an absolute value loss function). When working with small sample sizes (i.e., less than 50), the basic / reversed percentile and percentile confidence intervals for (for example) the variance statistic will be too narrow. So that with a sample of 20 points, 90% confidence interval will include the true variance only 78% of the time.[39] The basic / reverse percentile confidence intervals are easier to justify mathematically[40][37] but they are less accurate in general than percentile confidence intervals, and some authors discourage their use.[37]

• Studentized bootstrap. The studentized bootstrap, also called bootstrap-t, is computed analogously to the standard confidence interval, but replaces the quantiles from the normal or student approximation by the quantiles from the bootstrap distribution of the Student's t-test (see Davison and Hinkley 1997, equ. 5.7 p. 194 and Efron and Tibshirani 1993 equ 12.22, p. 160):

${\displaystyle (\theta -t_{(1-\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta },\theta -t_{(\alpha /2)}^{*}\cdot {\widehat {\text{se}}}_{\theta })}$ where ${\displaystyle t_{(1-\alpha /2)}^{*}}$ denotes the ${\displaystyle 1-\alpha /2}$ percentile of the bootstrapped Student's t-test ${\displaystyle t^{*}=({\widehat {\theta \,}}^{*}-{\widehat {\theta \,}})/{\widehat {\text{se}}}_{{\widehat {\theta \,}}^{*}}}$, and ${\displaystyle {\widehat {\text{se}}}_{\theta }}$ is the estimated standard error of the coefficient in the original model.

The studentized test enjoys optimal properties as the statistic that is bootstrapped is pivotal (i.e. it does not depend on nuisance parameters as the t-test follows asymptotically a N(0,1) distribution), unlike the percentile bootstrap.

• Bias-corrected bootstrap – adjusts for bias in the bootstrap distribution.
• Accelerated bootstrap – The bias-corrected and accelerated (BCa) bootstrap, by Efron (1987),[14] adjusts for both bias and skewness in the bootstrap distribution. This approach is accurate in a wide variety of settings, has reasonable computation requirements, and produces reasonably narrow intervals.

Efron and Tibshirani[1] suggest the following algorithm for comparing the means of two independent samples: Let ${\displaystyle x_{1},\ldots ,x_{n}}$ be a random sample from distribution F with sample mean ${\displaystyle {\bar {x}}}$ and sample variance ${\displaystyle \sigma _{x}^{2}}$. Let ${\displaystyle y_{1},\ldots ,y_{m}}$ be another, independent random sample from distribution G with mean ${\displaystyle {\bar {y}}}$ and variance ${\displaystyle \sigma _{y}^{2}}$

1. Calculate the test statistic ${\displaystyle t={\frac {{\bar {x}}-{\bar {y}}}{\sqrt {\sigma _{x}^{2}/n+\sigma _{y}^{2}/m}}}}$
2. Create two new data sets whose values are ${\displaystyle x_{i}^{'}=x_{i}-{\bar {x}}+{\bar {z}}}$ and ${\displaystyle y_{i}^{'}=y_{i}-{\bar {y}}+{\bar {z}},}$ where ${\displaystyle {\bar {z}}}$ is the mean of the combined sample.
3. Draw a random sample (${\displaystyle x_{i}^{*}}$) of size ${\displaystyle n}$ with replacement from ${\displaystyle x_{i}^{'}}$ and another random sample (${\displaystyle y_{i}^{*}}$) of size ${\displaystyle m}$ with replacement from ${\displaystyle y_{i}^{'}}$.
4. Calculate the test statistic ${\displaystyle t^{*}={\frac {{\bar {x^{*}}}-{\bar {y^{*}}}}{\sqrt {\sigma _{x}^{*2}/n+\sigma _{y}^{*2}/m}}}}$
5. Repeat 3 and 4 ${\displaystyle B}$ times (e.g. ${\displaystyle B=1000}$) to collect ${\displaystyle B}$ values of the test statistic.
6. Estimate the p-value as ${\displaystyle p={\frac {\sum _{i=1}^{B}I\{t_{i}^{*}\geq t\}}{B}}}$ where ${\displaystyle I({\text{condition}})=1}$ when condition is true and 0 otherwise.

In 1878, Simon Newcomb took observations on the speed of light.[41] 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 need exist for a heavy-tailed distribution.) A well-defined and robust statistic for 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. A convolution method of regularization reduces the discreteness of the bootstrap distribution by adding a small amount of N(0, σ²) random noise to each bootstrap sample. A conventional choice is ${\displaystyle \sigma =1/{\sqrt {n}}}$ for sample size n.

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.

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.

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 bootstrap resampling.

Bootstrap aggregating (bagging) is a meta-algorithm based on averaging the results of multiple bootstrap samples.

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.

• Statistics101: Resampling, Bootstrap, Monte Carlo Simulation program. Free program written in Java to run on any operating system.