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Various central limit theorems are of the form $a_n(\hat{\theta}-\theta)\sim N(0, \Sigma)$ approximately as $n \to \infty$ and usually $a_n = n^{1/2}$. Are there central limit theorems for statistics used in real applications where $a_n \neq n^{1/2}$?

cgmil
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    Yes, but typically they converge to other distributions. Research extreme value theory and the generalized extreme-value distribution, for instance. We also have a few examples of convergence to a Normal at slower rates: see https://stats.stackexchange.com/questions/203894 and https://stats.stackexchange.com/questions/406903 for instance. My answer in the latter thread provides a collection of examples leading to just about any desired asymptotic behavior. Please note that these theorems depend on more than just a statistic: you have to posit some kind of a sequence of random variables, too. – whuber May 19 '22 at 19:49
  • @whuber I'm aware of the extreme value results. The latter situation of converging to Normal at a slower rate is more what I'm asking about. Your first link suggests a situation more mathematically interesting than practical. The latter is better since, as in your discussion, there are situations where the estimators you discuss could be useful, although I'm unsure if you do manage to quantify the rate of convergence. – cgmil May 19 '22 at 21:09
  • I think I don't understand what you mean by "quantify," then, because at the end I point out "the plot of the SD against the sample size has a slope of $−p/2.$" I had thought that was pretty quantitative! – whuber May 19 '22 at 21:17
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    Also happens in nonparametrics, like density estimation... – kjetil b halvorsen May 19 '22 at 21:23
  • @whuber Actually on second reading, you are right; that's an actionable example. – cgmil May 19 '22 at 21:56

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Let an i.i.d. sample from a Normal random variable with density $$\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right).$$ Assume you specify a Skew Normal likelihood and attempt to estimate its parameters based on the above sample. The Skew Normal density is $$\frac{2}{s}\phi\left(\frac{x-\xi}{s}\right)\Phi\left(\lambda\frac{x-\xi}{s}\right)$$.

Then, in such a case where the true value of $\lambda =0$,

Chiogna, M. (2005). A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution. Statistical Methods and Applications, 14(3), 331-341,

has proven that

$$n^{1/6}\hat \lambda_{MLE} \rightarrow_d Z^{1/3},\;\;\; Z\sim N(0, V_z).$$

The interesting question here is, does that mean that

$$n^{1/2}\hat \lambda^3_{MLE} \rightarrow_d Z\;\;\; ???$$

Alecos Papadopoulos
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