For scale and shape , the log-logistic distribution has the displayed cumulative distribution function for , with density for . Its log-odds are affine in , giving the name and motivating a log-logistic probability plot. In the exam's parameterization, and .
For , plotting the logits of against sorted log observations gives an approximately straight line under the model. The probability integral transform makes a uniform order statistic with expectation , motivating these plotting positions. With those axes the slope is and the intercept is .
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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.