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

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.