**A**ware of my fascination for paradoxes (and heterodox publications), Ewan Cameron sent me the link to a recent arXival by Louis Lyons (Oxford) on different asymptotic distributions of the likelihood ratio. Which is full of approximations. The overall point of the note is hard to fathom… Unless it simply plans to illustrate Betteridge’s law of headlines, as suggested by Ewan.

For instance, the limiting distribution of the log-likelihood of an exponential sample at the true value of the parameter τ is not asymptotically Gaussian but almost surely infinite. While the log of the (Wilks) likelihood ratio at the true value of τ is truly (if asymptotically) a Χ² variable with one degree of freedom. That it is not a Gaussian is deemed a “paradox” by the author, explained by a cancellation of first order terms… Same thing again for the common Gaussian mean problem!

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