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I have just learned about the concept of bootstrapping, and a naive question came to mind: If we can always generate numerous bootstrap samples of our data, why bother to obtain more "real" data at all?

I do think I have an explanation, please tell me if I'm correct: I think the bootstrapping process reduces variance, BUT if my original dataset is BIASED, than I'm stuck with low variance and high bias, no matter how many replicas I'm taking.

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    $\begingroup$ bootstrapping doesn't create more information than is already in the data (and the model) ... actual data can give you more information $\endgroup$
    – Glen_b
    May 20, 2017 at 16:16
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    $\begingroup$ I agree with Glen_b that it does not create more information but I don't agree that it can give you less information. As I said in my answer it does not always work well but that can be said of any statistical method. $\endgroup$ May 20, 2017 at 16:35
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    $\begingroup$ Interesting question - perhaps a related concept is why does the bootstrap work?. Understanding this will help to know when it's useful. I thought of the bootstrap as an improvement over the normal approximation for sampling distributions. It can handle departures from normality that are not too extreme. It's other attractive feature is you don't need to do analytic/algebraic work - the replication does this for you. $\endgroup$ May 21, 2017 at 10:21

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The bootstrap is a method of doing inference in a way that does not require assuming a parametric form for the population distribution. It does not treat the original sample as if it is the population even those it involves sampling with replacement from the original sample. It assumes that sampling with replacement from the original sample of size n mimics taking a sample of size n from a larger population. It also has many variants such as the m out of n bootstrap which re-samples m time from a sample of size n where m < n. The nice properties of the bootstrap depend on asymptotic theory. As others have mentioned the bootstrap does not contain more information about the population than what is given in the original sample. For that reason it sometimes doesn't work well in small samples.

In my book "Bootstrap Methods: A Practitioners Guide" second edition published by Wiley in 2007, I point out situations where the bootstrap can fail. This includes distributions that do not have finite moments, small sample sizes, estimating extreme values from the distribution and estimating variance in survey sampling where the population size is N and a large sample n is taken. In some cases variants of the bootstrap can work better than the original approach. This happens with the m out of n bootstrap in some applications In the case of estimating error rates in discriminant analysis, the 632 bootstrap is an improvement over other methods including other bootstrap methods..

A reason for using it is that sometimes you can't rely on parametric assumptions and in some situations the bootstrap works better than other non-parametric methods. It can be applied to a wide variety of problems including nonlinear regression, classification, confidence interval estimation, bias estimation, adjustment of p-values and time series analysis to name a few.

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A bootstrap sample can only tell you things about the original sample, and won't give you any new information about the real population. It is simply a nonparametric method for constructing confidence intervals and similar.

If you want to gain more information about the population, you have to gather more data from the population.

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