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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!

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  • $\begingroup$ Related: stats.stackexchange.com/questions/9664/… $\endgroup$ – cardinal Mar 23 '12 at 16:15
  • $\begingroup$ @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! $\endgroup$ – Tianyang Li Mar 23 '12 at 16:32
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    $\begingroup$ Note that there is an open-access version of the Bickel and Freedman paper at projecteuclid. $\endgroup$ – cardinal Mar 23 '12 at 16:33
  • $\begingroup$ Also crossposted to math.SE: math.stackexchange.com/questions/123643/… $\endgroup$ – cardinal Mar 23 '12 at 16:36
  • $\begingroup$ @cardinal Sorry for that, I already corrected it. Do you know any reference that I can read on using parametric bootstrap for this problem? $\endgroup$ – Tianyang Li Mar 23 '12 at 16:54
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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.

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  • $\begingroup$ The Davidson and Hinkly text is from 1997 - no? Is there a link to a revised version? $\endgroup$ – B_Miner Mar 24 '12 at 14:39
  • $\begingroup$ 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. $\endgroup$ – 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

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