Log-Logistic Distribution
X ~ LogLogistic(α, β)
Proportional-Odds Survival Family • Published July 22, 2026
The first sheet to open the site's Lifetime/Survival category on its own terms: defined by exponentiating the already-published Logistic distribution (exact tie confirmed via Kolmogorov-Smirnov test), with a signature 'proportional odds' survival property and a hazard that can rise and then fall -- a shape neither the already-published Weibull nor Exponential can produce. Verified against scipy.stats.fisk to machine precision, with no closed-form maximum-likelihood estimator.
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About This Reference Sheet
The first sheet to open the site's Lifetime/Survival category on its own terms: defined by exponentiating the already-published Logistic distribution (exact tie confirmed via Kolmogorov-Smirnov test), with a signature 'proportional odds' survival property and a hazard that can rise and then fall -- a shape neither the already-published Weibull nor Exponential can produce. Verified against scipy.stats.fisk to machine precision, with no closed-form maximum-likelihood estimator.
Support
x ∈ (0, ∞)
Parameters
α > 0 (scale), β > 0 (shape)