# Intuition for Wilks' theorem

I'm trying to wrap my head around why it is intuitive that (under certain conditions) the likelihood ratio statistic follows a chi-squared distribution, asymptotically.

I've looked at the excellent answers to this and this question and can follow the derivations that use a Taylor expansion around the Maximum Likelihood Estimate and a Central Limit Theorem. However, once I step back, I don't find this result very intuitive.

I'm hoping there exists a heuristic, high-level argument for why the log of the ratio of the two likelihoods in question (or alternatively the difference of their logs) would asymptotically behave like (the sum of) one (or more independent) squared standard normal random variable(s).

• The LR is not "the ratio of ... two log likelihoods." Instead, it is the logarithm of the ratio. There's a huge difference!
– whuber
Nov 12, 2021 at 18:36
• @whuber great spot, sorry for the typo! Nov 12, 2021 at 18:40
• Okay, I'm glad that you have the definition correct--that makes this a useful question.
– whuber
Nov 12, 2021 at 19:09