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I already asked a question about the interpretation of the results of the bootstrap algorithm in case of a normal mixture. This time, I fit a hyperbolic distribution. According to this thread:

the use of standard errors in this family is completely unreliable since the distributions of the estimators are quite asymmetric as shown by the histogram of the bootstrap samples

To keep things short I will just give you the histogram plots. I am also using the alpha,beta,delta,mu parameterization.

Alpha

alpha

Beta

beta

Delta

delta

Mu

mu

From these samples of estimators I calculate the standard deviation and these give the standard errors of the alpha, beta, delta and mu estimates of the hyperbFit command.

Are the distributions of the estimators asymmetric? Are my result reliable?

I fit the hyperbolic distribution to financial return data.

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It seems to me that some quite different questions are bundled together here. In the thread Results of bootstrap reliable? I asked, in effect, for an explanation of what you mean by "reliable" and that request still stands. But I will subdivide the questions as I see them.

  • Are these results surprising? Without your data we cannot check, but viscerally, they are completely unsurprising to me. Fitting univariate distributions with several parameters is often very tricky, even with carefully chosen parameterisation and even if the data are a good candidate for the distribution being fitted. The territory here, with an example of financial returns data, is long-tailed, possibly skewed distributions where theoretical people have a great deal of fun thinking of new distributions but the results often disappoint people new to the field, because often no particular distribution is quite right for a particular dataset. The occurrence of outliers remains something to look out for.

  • Are the results trustworthy? Much of the point of bootstrapping I take as being to give a honest picture of the uncertainty possible with a given procedure and a given dataset, although bootstrapping like anything else has its limitations. In another thread, a member commented "I do not like the bootstrap method ... because it gave me too much different results". I recognise the feeling, but this is very much like not watching the news on television on the grounds that it is usually bad. So, without paradox, you can usually trust the bootstrap to signal if a procedure is untrustworthy in the sense that results can be very unstable.

  • How could the results be used? With such asymmetric distributions, you are quite right that estimates $\pm$ multiple of SEs would be poor choices for confidence intervals, but the answer is immediate: use percentile-based confidence intervals, or if needed and desired something more subtle as covered by any bootstrap text. This should come out of your software, but I am not familiar with the R functions you use.

A note of caution in this case: Financial returns I understand to be time series, but bootstrapping strict sense is based on an assumption of independent observations.

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  • $\begingroup$ +1 for your answer, but I think the distributions of my estimates is quite symmetric or not? Except the delta. I only used 1000 bootstraps and for this the results seem to be ok or not? How should they look in a perfect case? I mean isn't this result "ok"? The data is again the data given in the linked thread. I just now fitted a hyperbolic distribution and not a normal mixture. $\endgroup$ – Jen Bohold Aug 15 '13 at 8:30
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    $\begingroup$ I would not report any of these distributions as symmetric given the outliers and long tails. 1000 is a small sample size for bootstrapping these days. The perfect case would be assessed by simulating from a hyperbolic distribution, but you have four parameters and the shape of the sampling distributions is likely to depend on all of them. "OK" is no easier to answer than "reliable". $\endgroup$ – Nick Cox Aug 15 '13 at 8:38
  • $\begingroup$ Well, it's nice that you answer fast and very detailed, but it does not help me. Because for my thesis I need an evaluation. I said that the MLE converges slowly and the distributions of the estimates can be quite asymmetric. In my thesis I check this with the histograms. Since I have no experience with bootstrap I don't know what to conclude? So what should I conclude? I use these bootstraps to later one give the standard errors. $\endgroup$ – Jen Bohold Aug 15 '13 at 8:43
  • $\begingroup$ Sorry, but I think you are expecting more from me than I can give. Your results here must be part of a larger project. I can't possibly tell you what to conclude. In particular, the variability of the results has no direct bearing on whether this distribution fits well or indeed better than other distributions. I also think you are expecting more from Cross Validated than it can possibly give. People will give you a small amount of their time if they think your question is interesting; they are not offering complete consultancy support and cannot be proxy supervisors or advisers. $\endgroup$ – Nick Cox Aug 15 '13 at 9:04

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