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It is known that the bootstrap can fail.

I read in Section 6 of Bickel and Freedman (1981) that the bootstrap fails when you wan to use it to evaluate the MLE for estimating the parameter of a continuous uniform distribution.

I read Secion 7.4 of the book by Efron and Tibshirani but I'm not able to find the reference they pointed to.

Could someone point me to some more easily accessible things that I could refer to? Thanks!

conjugateprior
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Tianyang Li
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  • Related: http://stats.stackexchange.com/questions/9664/what-are-examples-where-a-naive-bootstrap-fails – cardinal Mar 23 '12 at 16:15
  • @cardinal I've also read that the parametric bootstrap could be used in this case to solve this problem to some extent. Can you please provide me with some reference? Thanks! – Tianyang Li Mar 23 '12 at 16:32
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    Note that there is an [open-access version](http://projecteuclid.org/euclid.aos/1176345637) of the Bickel and Freedman paper at projecteuclid. – cardinal Mar 23 '12 at 16:33
  • Also crossposted to math.SE: http://math.stackexchange.com/questions/123643/alternative-monte-carlo-methods-to-the-bootstrap-for-evaluating-estimators – cardinal Mar 23 '12 at 16:36
  • @cardinal Sorry for that, I already corrected it. Do you know any reference that I can read on using parametric bootstrap for this problem? – Tianyang Li Mar 23 '12 at 16:54
  • I'm a little confused here. Bootstrap seeks to replace an unknown distribution with its empirical distribution, and then some analytic results can be shown when the true distribution is assumed known. However, if you already *know* that your data are coming from a uniform, why wouldn't you just use Monte Carlo to get samples from that analytic uniform distribution? I might be misunderstanding your intention, but your question seems confused to me. – ely Mar 23 '12 at 17:07
  • To clarify, the Bickel/Freedman result says something more like "if it happened that the underlying distribution was continuous uniform, and here you were trying to get statistics about it from resampling its empirical distribution, then you'd be screwed." However, if you're already *assuming a model* in which the data are generated from a uniform, then getting parameter estimates from an MLE is a fine thing to do, even for the max of a uniform. – ely Mar 23 '12 at 17:09
  • @EMS This is a very good suggestion! I'm actually trying to solve a problem very like this one. – Tianyang Li Mar 23 '12 at 17:17
  • I see, so am I right to restate your questions as "What is a numerical sampling scheme that relies on the empirical sampling distribution, but which will get the right answer for the continuous uniform (where the usual bootstrap fails)?" ... and associated references? – ely Mar 23 '12 at 17:38
  • @EMS Yeah, I think that's more accurate, I've asked another question. – Tianyang Li Mar 24 '12 at 03:43

2 Answers2

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A good thorough review of bootstrap theory and applications is Davison and Hinkley, 1997. It's more up to date than your reference, goes a bit more gently, and has a lot of example (some of them in R). If that still looks too much, Mooney and Duval, 1993 is a simpler shorter introduction, and very good place to start.

Davison and Hinkley have a discussion of situations where bootstrapping fails at the end of ch. 2 (section 2.6). In fact an 'estimate the maximum' problem is in Example 2.5.

Unsurprisingly, in general the bootstrap fails when the empirical distribution function fails to stand in well for the real one. The specifics of failure -- concerning lack of approximate pivotally and edgeworth expansions -- are perhaps better left for the reading.

conjugateprior
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  • The Davidson and Hinkly text is from 1997 - no? Is there a link to a revised version? – B_Miner Mar 24 '12 at 14:39
  • I just checked the link and also my personal copy; 1997 it was. And no further editions so far as I can see. Post is now corrected. Thanks. – conjugateprior Mar 24 '12 at 14:45
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My book has a whole chapter on it (Chapter 9). The volume Exploring the Limits of Bootstrap is a conference proceeding that has research papers on it. Here are amazon links to these books.

Bootstrap Methods: A Guide for Practitioners and Researchers

Exploring the Limits of Bootstrap

Michael R. Chernick
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