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I have a data set; sample size is 16, the number of independent variable is 18 and one dependent variable . there are correlations between independent variables. I want to conduct Monte Carlo according the linear regression relationship of these variables but some of independent variables do not follow known distributions. in this regard, can anyone suggest a way ?

Thank you in advance

(I read this article article, but I have no experience with simulation)

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    $\begingroup$ "there are correlations between independent variables" does not sound right, does it?! $\endgroup$
    – Xi'an
    Oct 26, 2015 at 8:17
  • $\begingroup$ I edited your question to change [correlation] tag into [bootstrap] as it seems to describe your question better. $\endgroup$
    – Tim
    Oct 26, 2015 at 9:03

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Ugh, by a cursory look at the article you linked, what he describes is known in the statistics community as the Bootstrap. In particular the author first mentions the shortcoming of the parametric bootstrap, where you draw each of the covariates independently of the others. And then he proposes a remedy to account for the correlations and simulate while taking these into account. I think in your case this will not work because your estimate of the correlation will also be very bad!! (18 variables and only 16 samples)

A lot simpler solution is to just use the empirical bootstrap, which automatically preserves such correlations. This means, that to do your "monte carlo" experiment you preceed as follows within each replication:

Out of your 16 samples, draw 16 with replacement. Then just take these rows of your data corresponding to these samples (so some rows, each with the 18+1 variables, will be replicated!) and do whatever analysis you want for them. Summarize the result.

Since you are a in a regression setup, there also exist more complicated methods, such as the residual and the wild bootstrap, but these might be too complicated for now.

But basically googling for Bootstrap (statistic) will give you a lot more answers than this confusing paper.

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