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

  • $\begingroup$ 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
  • 2
    $\begingroup$ "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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.