I am drawing numbers from a Gamma distribution with shape and scale parameters $k$ and $\theta$, respectively. I, then take the square root of every number drawn.
It sounds like a waste of computational time to compute those square roots. Instead of taking this square root everytime, what distribution could I sample from? For example, I don't think that drawing numbers from a Gamma distribution with shape and scale parameters $\sqrt{k}$ and $\sqrt{\theta}$ would be equivalent?
Best Answer
It's called the Nakagami distribution. If $Y\sim \mathrm{Gamma}(k, \theta)$, then $X=\sqrt{Y}$ is distributed via $$ f(x) = \frac{2}{\Gamma(k)\theta^k}x^{2k-1} e^{-x^2/\theta} $$ Alternatively, you can first sample $Z$ from a Chi distribution with paramater $2k$, and then scale it as $X=\sqrt{\theta/2}\:Z$. This gives the same distribution. Scaling is much faster than square root, but would guess you don't even need to scale. Just remember all your computations are done in units of $\sqrt{\theta/2}$ (in other words, $\theta$ is somewhat irrelevant for sampling purposes).