Can someone please point me to a textbook or lecture notes that explains what variance stabilising transformations are? I can only find bits and pieces on google.
I don't know a lot of statistics, but I know a decent amount of probability theory.
2 Answers
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There is a quite brief introduction to variance-stabilising transformations in Asymptotic Statistics by A.W. van der Vaart (2000). Many references to variance-stabilising transformations in textbooks/lecture notes will be in the context of the delta method.
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A nice, although short, introduction are given in these lecture notes.
The basic idea in the setting of sample means, e.g. $X_i$ iid with mean $\mu$ and variance $\sigma(\mu)^2$, is to use CLT and delta method to find a function $g$ such that $$ n^{1/2}\left( g(\bar{X}_n) - g(\mu)\right) \overset{d}{\to}N(0, g'(\mu)^2\sigma(\mu)^2), $$
and $g'(\mu)\sigma(\mu)=c$ for some $c\in\mathbb R$. This is just an example, of course. The idea is the same if you consider more general convergence in distribution and transformations $g$.
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$\begingroup$ your statement $c \in \mathbb{R}$ should be qualified as also non-negative $c>0$. Negative variance makes no sense. $\endgroup$ Commented Sep 10, 2018 at 23:19
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1$\begingroup$ I have fixed a typo that caused the confusion @LucasRoberts. $\endgroup$– KOECommented Sep 12, 2018 at 13:23