Can somebody identify this distribution?

Question

I am searching for the name of the distribution associated with this density on $\mathbb{R}_+$:

$$p(r|\lambda) = \frac{2\lambda r\exp\left(\lambda\exp\left(-r^{2}\right)-r^{2}\right)}{\exp\left(\lambda\right)-1}.$$

It arises from the mixture distribution of
$$p(r|n)=2nr\cdot\exp\left(-nr^{2}\right)$$

($n\in \mathbb N_+$) with mixture weights from a "positive" Poisson distribution on $n>0$ $$p(n|\lambda)=\frac{\lambda^n \exp(-\lambda)}{n! (1-\exp(-\lambda))}$$

I first thought it looked like a Gumbel distribution, but I couldn't get it into the right shape.

Answer 1

I think it relates to a Gumbel distribution:

if I define $S=R^2$, when $R\sim p(r|\lambda)$, the density of $S$ is given by the Jacobian formula: \begin{align*} q(s|\lambda) &= p(\sqrt{s}|\lambda)\times \left|\frac{\text{d}r}{\text{d}s}\right|\\ &= p(\sqrt{s}|\lambda)\times \frac{1}{2\sqrt{s}}\\ &= \frac{2\lambda \sqrt{s}\exp\left(\lambda\exp\left(-s\right)-s\right)}{\exp\left(\lambda\right)-1}\,\frac{1}{2\sqrt{s}}\\ &= \frac{\lambda \exp\left(\lambda\exp\left(-s\right)-s\right)}{\exp\left(\lambda\right)-1}\\ &= \frac{\exp\left(\exp\left(\log\{\lambda\}-s\right)+\log\{\lambda\}-s\right)}{\exp\left(\lambda\right)-1}\\ &= \frac{\exp\left(\exp\left(-z\right)-z\right)}{\exp\left(\lambda\right)-1}\\ \end{align*} where $z=s-\log\{\lambda\}$

So $S$ is almost distributed as a Gumbel distribution with parameter $(\log\{\lambda\},1)$ except that (a) its support is truncated to $(0,+\infty)$ and (b) there is a missing - in front of the exponential inside the exponential...

This means that $S-\log\{\lambda\}$ has a fixed distribution with the above density and cdf $F$. From this representation, there exists a transform $G^{-1}\circ F$ (with $G$ being the cdf of the Gumbel distribution) that turns $ S-\log\{\lambda\}$ into a standard Gumbel, but this is not very useful!

Note that, in the mixture representation, $S=R^2$ is then distributed as an Exponential $\mathcal{E}(n)$ variate.