# VST for negative binomial random variable

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?

• 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.
– whuber
Commented Apr 14, 2022 at 21:50
• I am looking for the best vst. Commented Apr 14, 2022 at 22:00
• "Best" in what sense?
– whuber
Commented Apr 14, 2022 at 22:01
• In the sense that the Anscombe transform is better than the sqrt transform for Poison distribution. Commented Apr 15, 2022 at 3:33

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.