Is there a name for $(\mathrm{e}^{x^2/{2\sigma^2}}-\lambda)^{-1}$, $0 \leq \lambda < 1$?

Question

Is there a name given to a distribution of the form:

$(\mathrm{e}^{x^2/{2\sigma^2}}-\lambda)^{-1}$, where $0 \leq \lambda < 1$?

I see that for $\lambda=0$, this is just the normal distribution with variance $\sigma^2$, up to a scale factor.

Answer 1

The required normalization constant for the specified two-parameter likelihood is $$\frac{\lambda}{\sqrt{2\pi} \sigma \operatorname{Li}_{1/2}(\lambda)},$$ where $$\operatorname{Li}_n(z) = \sum_{k=1}^\infty \frac{z^k}{k^n}$$ is the polylogarithm function. In the limiting case where $\lambda \to 0$, this reduces to $1/(\sqrt{2\pi} \sigma)$ as expected. Thus the density of such a random variable would be $$f_X(x) = \begin{cases} \displaystyle \frac{\lambda}{\sqrt{2\pi} \sigma \operatorname{Li}_{1/2}(\lambda) (e^{x^2/(2\sigma^2)} - \lambda)}, & 0 < \lambda < 1 \\ \displaystyle \frac{1}{\sqrt{2\pi \sigma}} e^{-x^2/(2\sigma^2)}, & \lambda = 0. \end{cases}$$ To my knowledge I have not encountered such a distribution before.

Qualitatively, as $\lambda \to 1$ for a fixed $\sigma$, the density gets more sharply peaked at $0$.