The linked sites (link1, link2) demonstrate that the likelihood ratio tests and the corresponding one- and two-sample t-tests are equivalent. However, based on my understanding, the null distribution of the likelihood ratio test (i.e., chi-square distribution) is an approximation, whereas the t-distribution in the t-tests is not. Despite this difference, why are they equivalent?
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You need to distinguish two different things:
With a likelihood ratio statistic, $\Lambda$, the chi-squared distribution is an asymptotic approximation to the distribution of $-2\log(\Lambda)$ under quite broad conditions.
Under the conditions where some t-test is exact, the corresponding likelihood ratio statistic might be shown to be monotonic in $|t|$ or $t^2$ (whichever is convenient), and therefore the two statistics should lead to equivalent tests, if the correct null distribution for that statistic is used; it won't be distributed as chi-squared in small samples.
Note in particular, that the square of a $t$-distributed random variable has an $F$ distribution, and that asymptotically as the denominator d.f. goes to infinity, the $F$ distribution goes to a scaled chi-squared. You may be able to derive an equivalent statistic to $-2\log(\Lambda)$ that has a small-sample $F$ distribution and in the limit as $n$ goes to infinity that $F$ will be proportional to a chi-squared with the same d.f. as the numerator d.f. of the F statistic (and indeed the same d.f. as the $t$).
