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https://en.wikipedia.org/wiki/Variance-stabilizing_transformation

The variance stabilizing transform of a Poisson random variable is the Anscombe transform. What is the VST for a negative binomial random variable?

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  • $\begingroup$ Re "The:" a variance stabilizing transform is not unique. For instance, the square root is another variance-stabilizing transform of the Poisson family. A general method for finding a variance-stabilizing transform is given at stats.stackexchange.com/questions/251449. $\endgroup$
    – whuber
    Commented Apr 14, 2022 at 21:50
  • $\begingroup$ I am looking for the best vst. $\endgroup$ Commented Apr 14, 2022 at 22:00
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    $\begingroup$ "Best" in what sense? $\endgroup$
    – whuber
    Commented Apr 14, 2022 at 22:01
  • $\begingroup$ In the sense that the Anscombe transform is better than the sqrt transform for Poison distribution. $\endgroup$ Commented Apr 15, 2022 at 3:33

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Take a look at Anscombe's paper from 1948 the transformation of Poisson, Binomial and Negative-Binomial data. There, he shows that for a random variable $$ Y \sim \text{NegBin}(m, k), $$ where $m$ is the mean and $k$ is the size, the transformation $$ Z = \sinh^{-1}\sqrt{\frac{Y + c}{k - 2c}} $$ with $c \approx 3/8$ is in some sense optimal.

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