 Log-logistic distribution

For a scale $a>0$ and shape $\kappa >0$, the log-logistic distribution has cumulative distribution function and probability density functionIt has no point mass at zero. Its quantile function is $Q(p)=a[p/(1-p){]}^{1/\kappa }$ for $0<p<1$, so its median is $a$. The log odds relation $\log (F(x)/(1-F(x)))=\kappa \log x-\kappa \log a$ explains both its name and the log-logistic probability plot.^[ref]

Its survival function is $1/(1+(x/a{)}^{\kappa })$, with power tail proportional to $x^{-\kappa }$. A positive moment $\mathbb{E}{X}^r$ is finite exactly when $r<\kappa$: near infinity its defining integral behaves like $\int x^{r-\kappa -1}dx$. For example, $a=2,\kappa =2$ gives median $2$ and $\mathbb{P}(X>4)=1/5$. Writing $a=\alpha$ and $\kappa =\beta -1$ is an alternative parameterization requiring $\beta >1$.

The log-logistic distribution is a continuous probability distribution used in statistics and reliability analysis. It is particularly useful for modeling the distribution of positive random variables, especially in contexts where the data exhibits a skewed distribution and has a long right tail. The log-logistic distribution is often employed in survival analysis and economics. ### Definition: A random variable \(X\) follows a log-logistic distribution if its logarithm, \(\log(X)\), follows a logistic distribution.