Statistics – What is the Inverse CDF / PPF of the Logit-Normal Distribution?

Question

In this post, I am trying to implement the logit-normal distribution in Python. The provided answer works for me, however, the rvs method that draws random variates failes for me. According to the documentation of the pdf class that I am using:

"The default method _rvs relies on the inverse of the cdf, _ppf, applied to a uniform random variate. In order to generate random variates efficiently, either the default _ppf needs to be overwritten (e.g. if the inverse cdf can expressed in an explicit form) or a sampling method needs to be implemented in a custom _rvs method."

This is what I am trying to figure out, but I couldn't find a description of the inverse logit-normal cdf anywhere. How do I do this?

Answer 1

I assume you want to generate $X$ where $Y=\log_e\left(\frac{X}{1-X}\right)\sim \mathcal N(\mu,\sigma^2)$

So generate $Y$, either directly from a normal distribution, or from a $U \sim \mathcal U(0,1)$ and $Y = \mu +\sigma\Phi^{-1}(U)$, and then let $X=\frac{e^Y}{e^Y+1}=\frac{1}{1+e^{-Y}}$. Combine these and you could say $$X=\frac{1}{1+e^{-\mu -\sigma\Phi^{-1}(U)}}$$ and in a sense this is the inverse of the cumulative distribution function of the logit-normal distribution