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Hello and thank you for taking the time.

I'm performing an LRT for a likelihood distribution which violates the regularity conditions for Wilks theorem and wald intervals. I'm running a monte-carlo where I generate small-sample size data(<20) from a rare physics process and luckily have access to run this on a high throughput computing service. The crux of this post is, if I ran the monte-carlo for several thousand iterations, can I brute force the relationship between test-statistic and p-value in a valid way? Since I know the true parameters from the monte-carlo, and the MLE from the LRT, if I take the difference in log-likelihood for the two, and repeat this many times can I use this distribution in place of the Chi-square distribution?

I'm approaching the problem this way because the wikipedia page for Wilks' Theorem says "To be clear: These limitations on Wilks’ theorem do not negate any power properties of a particular likelihood ratio test. The only issue is that a \chi ^{2} distribution is sometimes not appropriate for determining the statistical significance of the result. ". If I'm interpreting this correctly, is there a better(read as: more citeable) source that you know of?

Thanks for your help.

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  • $\begingroup$ Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69 (3), 683–734. I believe this suggests simulation to find the null distribution. $\endgroup$ – steveo'america May 28 at 19:15

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