# AN INTRODUCTION TO THE BOOTSTRAP PDF

57 An Introduction to the Bootstrap B. Efron and R. Tibshirani (). (Full details concerning this series are available from the Publishers.) An. Introduction. An introduction to bootstrap. Home ยท An introduction to bootstrap An Introduction to Confucianism (Introduction to Religion). Read more. ๐ฃ๐๐ | On Jan 1, , A. C. Davison and others published An Introduction to the Bootstrap with Applications in R.

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Introduction to the. Bootstrap. Bradley Efron. Department of Statistics. Stanford University and. Robert J. Tibshirani. Department of Preventative Medicine and. Library of Congress Cataloging-in-Publication Data. Efron, Bradley. An introduction to the bootstrap /Brad Efron, Rob Tibshirani. p. cm. Includes bibliographical. Introduction to the Bootstrap. 1 Motivation. The traditional approach to statistical inference relies on idealized models and assumptions.

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.

## An Introduction to the Bootstrap Method

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.

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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". Types of bootstrap scheme[ edit ] This section includes a list of references , related reading or external links , but 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 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.

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For other problems, a smooth bootstrap will likely be preferred. But some embed codes will be used as a concept illustrating. We will do a introduction of Bootstrap resampling method, then illustrate the motivation of Bootstrap when it was introduced by Bradley Efron , and illustrate the general idea about bootstrap.

Related Fundamental knowledge The ideas behind bootstrap, in fact, are containing so many statistic topics that needs to be concerned. However, it is a good chance to recap some statistic inference concepts!

## An introduction to bootstrap

Some ideas may cover with advance statistic, but I will use a simple way and not very formal mathematics expressions to illustrate basic idea as simple as I can. Links at the end of the article will be provided if you want to learn more about these concepts. It is a resampling method by independently sampling with replacement from an existing sample data with same sample size n, and performing inference among these resampled data.

Generally, bootstrap involves the following steps: A sample from population with sample size n. Draw a sample from the original sample data with replacement with size n, and replicate B times, each re-sampled sample is called a Bootstrap Sample, and there will totally B Bootstrap Samples. We can see we generate new data points by re-sampling from an existing sample, and make inference just based on these new data points.

How and why does bootstrap work? Why use the simulation technique? In other word, how can I find a estimated variance of statistic by resampling? When Efron introduced the method, it was particularly motivated by evaluating of the accuracy of an estimator in the field of statistic inference.

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Usually, estimated standard error are an first step toward thinking critically about the accuracy statistical estimates. Scenario Case Imagine that you want to summarize how many times a day do students pick up their smartphone in your lab with totally students.

It's hard to summarize the number of pickups in whole lab like a census way. Instead, you make a online survey which also provided the pickup-counting APP.

## An introduction to bootstrap

In the next few days, you receive 30 students responses with their number of pickups in a given day. You calculated the mean of these 30 pickups and got an estimate for pickups is Codes for this case, just feel free to check out. In statistic field, the process above is called a point estimate. What we would like to know is the true number of pickups in whole lab.

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Population Parameter: Numeric summary about a population. One key question is โ How accurate is this estimate result? Hence, besides reporting the value of a point estimate, some indication about the precision should be given. The common measure of accuracy is the standard error of the estimate. It tells us how far your sample estimate deviates from the actual parameter.

If the standard error itself involves unknown parameters, we used the estimated standard error by replacing the unknown parameters with an estimate of the parameters.

In our case, our estimator is sample mean, and for sample mean and nearly only one! Standard Error in Statistic Inference Now we have got our estimated standard error. How can the standard error be used in the statistic inference? However, how this inference was going well is under some rigorous assumptions.

In our case is the approximated normal distribution. The standard error of a estimate is hard to evaluate in general. Configuration of the Bootstrap There are two parameters that must be chosen when performing the bootstrap: the size of the sample and the number of repetitions of the procedure to perform. Sample Size In machine learning, it is common to use a sample size that is the same as the original dataset.

The bootstrap sample is the same size as the original dataset.

As a result, some samples will be represented multiple times in the bootstrap sample while others will not be selected at all. Repetitions The number of repetitions must be large enough to ensure that meaningful statistics, such as the mean, standard deviation, and standard error can be calculated on the sample.

A minimum might be 20 or 30 repetitions. Smaller values can be used will further add variance to the statistics calculated on the sample of estimated values.

Ideally, the sample of estimates would be as large as possible given the time resources, with hundreds or thousands of repeats. Worked Example We can make the bootstrap procedure concrete with a small worked example. We will work through one iteration of the procedure.

Imagine we have a dataset with 6 observations: [0. Here, we will use 4.Same with previous Simulation part for simulating Var M. Importantly, any data preparation prior to fitting the model or tuning of the hyperparameter of the model must occur within the for-loop on the data sample.

In our case, we have sample with 30, and sample mean is In this paper we focus on the basic non-parametric bootstrap, and propose a vectorized implementation for statistics based on sample moments. 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. With the aid of computer, we can make B as large as we like to approximate to the sampling distribution of statistic M.

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