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. In this case a parametric model is fitted to the data, 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 quantity, or estimate, of interest is calculated from these data. This sampling process is repeated many times as for other bootstrap methods. 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. 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 to fit a probabilistic model from which replicates may then be drawn. Gaussian processes are methods from Bayesian non-parametric statistics but are here used to construct a parametric bootstrap approach, which implicitly allows the time-dependence of the data to be taken into account. The Wild bootstrap, proposed originally by Wu ,  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. This method assumes that the 'true' residual distribution is symmetric and can offer advantages over simple residual sampling for smaller sample sizes.
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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 instead blocks of data. The block bootstrap has been used mainly with data correlated in time i.
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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. 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.
Weighted bootstrap for U-statistics - ScienceDirect
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. 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.
The bootstrap distribution of a point estimator of a population parameter has been used to produce a bootstrapped confidence interval for the parameter's true value, if the parameter can be written as a function of the population's distribution. Population parameters are estimated with many point estimators.
Popular families of point-estimators include mean-unbiased minimum-variance estimators , median-unbiased estimators , Bayesian estimators for example, the posterior distribution 's mode , median , mean , and maximum-likelihood estimators. A Bayesian point estimator and a maximum-likelihood estimator have good performance when the sample size is infinite, according to asymptotic theory. For practical problems with finite samples, other estimators may be preferable.
Asymptotic theory suggests techniques that often improve the performance of bootstrapped estimators; the bootstrapping of a maximum-likelihood estimator may often be improved using transformations related to pivotal quantities. The bootstrap distribution of a parameter-estimator has been used to calculate confidence intervals for its population-parameter.
There are several methods for constructing confidence intervals from the bootstrap distribution of a real parameter:. See Davison and Hinkley , equ. 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  and where the sample statistic is median-unbiased and has maximum concentration or minimum risk with respect to an absolute value loss function.
In other cases, the percentile bootstrap can be too narrow. The studentized test enjoys optimal properties as the statistic that is bootstrapped is pivotal i. In , Simon Newcomb took observations on the speed of light. Note that 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. 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. 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. From Wikipedia, the free encyclopedia. This section includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations. Please help to improve this section by introducing more precise citations.
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Mathematica Journal , 9, Another look at the jackknife". The Annals of Statistics. Journal of the American Statistical Association. Journal of the American Statistical Association, Vol.
Does PLS have advantages for small sample size or non-normal data? MIS Quarterly, 36 3 , Resampling methods of estimation. Advising on research methods: Johannes van Kessel Publishing. Annals of Statistics , 9, Journal of the American Statistical Association, 89, Bootstrap methods and their application. Cambridge Series in Statistical and Probabilistic Mathematics. The jackknife, the bootstrap, and other resampling plans. Design and Analysis of Ecological Experiments.
Mathematical Statistics and Data Analysis 2 ed. December First available in Project Euclid: Permanent link to this document https: Zentralblatt MATH identifier Sampling theory, sample surveys Secondary: Bootstrap, jackknife and other resampling methods. Keywords Bootstrap bootstrap weights confidence intervals imputation missing data multistage designs pseudo-population approach survey sampling unequal probability sampling variance estimation.
A survey of bootstrap methods in finite population sampling. More by Zeinab Mashreghi Search this author in: Google Scholar Project Euclid. More by David Haziza Search this author in: Abstract Article info and citation First page References Abstract We review bootstrap methods in the context of survey data where the effect of the sampling design on the variability of estimators has to be taken into account. Article information Source Statist.