In teachingApps: Apps for Teaching Statistics, R Programming, and Shiny App Development

Functional relationships for $Y \sim LOGIS(\mu,\sigma),\;\;Y\in(-\infty,\infty)$

$$ \begin{aligned} f(y|\mu,\sigma)&=\frac{1}{\sigma}\phi_{logis}\left(\frac{y-\mu}{\sigma}\right)=\frac{1}{\sigma}\exp\left(\frac{y-\mu}{\sigma}\right)\left[1+\exp\left(\frac{y-\mu}{\sigma}\right)\right]^{-2}\\\\ F(y|\mu,\sigma)&=\Phi_{logis}\left(\frac{y-\mu}{\sigma}\right)=\exp\left(\frac{y-\mu}{\sigma}\right)\left[1+\exp\left(\frac{y-\mu}{\sigma}\right)\right]^{-1}\\\\ h(y|\mu,\sigma)&=\frac{1}{\sigma}\Phi_{logis}\left(\frac{y-\mu}{\sigma}\right)\\\\ y_{p}&=\mu+\Phi^{-1}{logis}(p)\sigma, \;\;\;\;\;\;\;\;\text{where}\;\Phi^{-1}{logis}(p)=\log [p/(1-p)]\\\\ E[Y]&=\mu\\\\ Var[Y]&=\sigma^2\pi^2/3 \end{aligned} $$

teachingApps documentation built on July 1, 2020, 5:58 p.m.