What positive distribution has the form $x^{-\ln(x)}$

Question

What is the name for the continuous distribution $f(x)\propto x^{-\ln(x)}$ for $x\geq 0$? More precisely, I'd like the name for the family of distributions of the form $f(x)\propto x^{-c \ln(x)}$, where $c$ is a positive constant.

As for the significance of this distribution, it's the simplest one that dies (approaches 0) at a super-polynomial and sub-exponential rate both as $x\to\infty$ and $x\to 0$. Also, fwiw, the normalizing constant for the distribution has a simple closed form: $$\sqrt{\frac{e^{c/2} \pi}{c}}$$

Answer 1

like in Does the distribution whose pdf is proportional to $\exp(-\log^2(x))$ have a name?,

\begin{align} x^{-\log x} &= \exp(-\log^2 x) \\ &\propto \exp(-\log^2 x)/\exp(0.25) \\ &= \exp(-\log^2 x - 0.25) \\ &= \exp(-(\log^2 x + 0.25)) \\ &= \exp(-(\log^2 x + 0.25 - \log x + \log x)) \\ &= \exp(-((\log^2 x - 0.5)^2 + \log x)) \\ &= \exp(-(\log^2 x - 0.5)^2 - \log x) \\ &= \exp(-(\log^2 x - 0.5)^2)/x \\ &propto \exp(-(\log^2 x - 0.5)^2)/x/\sqrt\pi \\ \end{align} This is a log-normal with $\mu=0.5$ and $\sigma = \sqrt{1/2}$. if there is a denominator for $\log(x)$ in the exponent, you can choose $\mu=\sigma^2$.

would've made a comment but i can't comment.

also numerically, where my distribution is exp(-log(x)^2) / sqrt(sqrt(ℯ)) / sqrt(π)

using Distributions
using Plots
inspectdr()
lw = 3
f(x) = exp(-log(x)^2) / sqrt(sqrt(ℯ)) / sqrt(π)
p = plot(f, label="my distribution", linestyle=:dot, linewidth=lw*2, size=(1000, 700))
dist = LogNormal(0.5, sqrt(0.5))
p = plot(p, x->pdf(dist, x), label="log normal (0.5, sqrt(0.5)", linestyle=:dash, linewidth=lw)
savefig(p, "fig.png")
p