Variance Stabilizing Transformations

I want to record here a very interesting thing which I recently discovered, variance-stabilizing transformations. The idea is very simple: suppose we have a random variable \(x\), which follows a probability distribution which is parametrized solely by its mean \(\mu\), with variance \(var(x) = g(\mu)\) a known function of the mean.

Now, suppose we take the new random variable \(y = f(x)\), for some yet-to-be-determined function \(f\). Assuming that \(x\) is reasonably localized about its mean, we can make the approximation

\( f(x) = f(\mu + (x-\mu)) \approx f(\mu) + f'(\mu)(x-\mu). \)

Then we have

\[ \begin{aligned} E[y] &\approx f(\mu) \cr var(y) &\approx f'(\mu)^2 var(x) \cr &= f'(\mu)^2 g(\mu) \end{aligned} \]

Now, suppose we want to choose \(f(x)\) so that \(var(y) \approx 1\). Using the above approximation, we have \( f'(\mu) = \frac{1}{\sqrt{g(\mu)}} \), which we can solve as

\( f(\mu) = \int \frac{1}{\sqrt{g(\mu)}} d\mu\)

Now let’s take the special case of the Poisson distrubution, where \(var(x) = \mu\). Then \(f(\mu) = \int \frac{1}{\sqrt \mu} d\mu = 2\sqrt{\mu}\). This is very nearly the Anscombe transform \(x \mapsto 2\sqrt{x+3/8}\). The additional shift by \(3/8\) can be understood by doing a more careful analysis of the variance under the transformation.