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Let $X \sim \text{Normal}(\mu, \sigma^2)$. Define $Y = \frac{e^X -1}{e^X+1}$. The inverse transformation is $X = \text{logit}\left(\frac{1+Y}{2}\right) = \log\left(\frac{1+Y}{1-Y} \right)$. By the transformation theorem $$ f_Y(y) = f_X\left[ \log\left(\frac{1+y}{1-y}\right) \right]\times\frac{2}{(1-y)(1+y)}. $$ Does this distribution have a name that I can look up? I have to evaluate this density pretty often when I use random walk Metropolis-Hastings and I sample for parameters $-1 < Y < 1$ (e.g. correlation parameters, AR(1) parameters, etc.) by transforming them into $X$ first, and then adding Gaussian noise to them.

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Thanks to @whuber's comment, we know if $X \sim \text{Normal}(\mu, \sigma^2)$, then $Z = e^x/(1+e^x)$ follows a $\text{Logit-Normal}(\mu, \sigma)$ and has density $$ f_Z(z) = \frac{1}{\sigma\sqrt{2\pi}}\frac{1}{z(1-z)}\exp\left[-\frac{(\text{logit}(z) - \mu)^2}{2\sigma^2}\right]. $$

Then $Y = \frac{e^X -1}{e^X+1} = 2\left(\frac{e^X}{1+e^X}\right)-1 = 2Z-1$ is just a scaled and shifted logit-normal random variable with density \begin{align*} f_Y(y) &= f_Z\left(\frac{y+1}{2}\right)\times \frac{1}{2} \\ &= \frac{1}{\sigma\sqrt{2\pi}}\frac{2}{(1+y)(1-y)}\exp\left[-\frac{\left\{\log\left(\frac{1+y}{1-y}\right) - \mu\right\}^2}{2\sigma^2}\right]\\ &= f_X\left[ \log\left(\frac{1+y}{1-y}\right) \right]\times\frac{2}{(1-y)(1+y)}. \end{align*} Not part of the same family, but still good to know.

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    $\begingroup$ +1. However, I would suggest that it is widely understood that most families of distributions extend naturally to location-scale families in this manner--especially families of continuous ones. After all, the distinction between $Y$ and $Z$ is merely a matter of what units you use to express the values of these variables. $\endgroup$ – whuber Jan 7 '18 at 15:58

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