There are some good reasons for this. The Central Limit Theorem is usually thrown in there as a justification, and it works reasonably well for practical applications. But the Central Limit Theorem really counts on that convergence as n goes to infinity. And it turns out that we can actually use this to our advantage using a method called the Bootstrap.
History[ edit ] The bootstrap was published by Bradley Efron in "Bootstrap methods: As the population is unknown, the true error in a sample statistic against its population value is unknown. 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 N 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. This process is repeated a large number of times typically 1, or 10, timesand for each of these bootstrap samples we compute its mean each of these are called bootstrap estimates.
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[ edit ] This section includes a list of referencesrelated reading or external linksbut its sources remain unclear because it lacks inline citations. Please help to improve this section by introducing more precise citations.
June Advantages[ edit ] 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 complex parameters 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.
The apparent simplicity may conceal the fact that important assumptions are being made when undertaking the bootstrap analysis e. Recommendations[ edit ] 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 lead to negligible improvements in the estimation of standard errors. 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 distributionthen 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 tailedone should hesitate to use the naive bootstrap".
Types of bootstrap scheme[ edit ] This section includes a list of referencesrelated reading or external linksbut its sources remain unclear because it lacks inline citations.
June Learn how and when to remove this template message In univariate problems, it is usually acceptable to resample the individual observations with replacement "case resampling" below unlike subsamplingin 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.
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. There are at least two ways of performing case resampling.A bootstrap sample is a smaller sample that is “bootstrapped” from a larger sample.
Bootstrapping is a type of re sampling where large numbers of smaller samples of the same size are repeatedly drawn, with replacement, from a single original sample. Bootstrapping is a resampling technique used to obtain estimates of summary statistics.
Business [ edit ] Bootstrapping in business means starting a business without external help or capital. Bootstrapping is a statistical technique that falls under the broader heading of resampling. This technique involves a relatively simple procedure but repeated so many times that it is heavily dependent upon computer calculations.
Bootstrapping provides a method other than confidence intervals to estimate a population parameter. Bootstrapping has yet another meaning in the context of reinforcement learning that may be useful to know for developers, in addition to its use in software development (most answers here, e.g.
by kdgregory) and its use in statistics as discussed by Dirk Eddelbuettel. create a large number of “phantom samples” known as bootstrap samples. The sample summary is then computed on each of the bootstrap samples (usually a few thousand).
A histogram of the set of these computed values is referred to as the bootstrap distribution of the statistic. Bootstrapping Statistics & Confidence Intervals, Tutorial If you use applied statistics in your career, odds are you’ve used the Great Assumption Of Our Era, the assumption of the Normal distribution.