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I know that the likelihood ratio test statistic follows a chi-squared distribution. But is this only asymptotically? I.e. that I have to have a large sample size in order for the test to be reliable? In smaller sample sizes, am I better off using an exact method?

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That depends on other assumptions holding or not.

If, for example, you are confident to be sampling from a normal distribution, then the $\chi^2$ distribution is exact.

If you do not wish to make such an assumption, then the $\chi^2$ distribution can only be established asymptotically.

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