Home / Statistical Tools / Distribution Fit/Calc / Distribution Fit / LogLogistic Distribution
LogLogistic Distribution (2-Parameter and 3-Parameter)¶
Distribution Fitting: Non-Reliability¶
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QXL Stat Tools Tab > Distribution Fit/Calc > Distribution Fit > 2-Param LogLogistic
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QXL Stat Tools Tab > Distribution Fit/Calc > Distribution Fit > 3-Param LogLogistic
Distribution Fitting: Reliability¶
Uncensored Data
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QXL Stat Tools Tab > Reliability > Uncensored > 2-Param LogLogistic
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QXL Stat Tools Tab > Reliability > Uncensored > 3-Param LogLogistic
Probability Calculator¶
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QXL Stat Tools Tab > Distribution Fit/Calc > Distribution Calculator > LogLogistic
The LogLogistic distribution is a continuous probability distribution also called the Fisk distribution. If a random variable X is LogLogistic, then Ln(X) will follow the Logistic distribution. The LogLogistic distribution can be 2-Parameter and 3-Parameter. If the offset (threshold) is zero, then the 2-- and 3-parameter are identical.
| Parameters | \(-\infty < \mu < +\infty \quad \mu = \text{Shape, Location}\) \(\sigma > 0 \quad \sigma = \text{Scale}\) \(-\infty < \gamma < +\infty \quad \gamma = \text{Offset (Threshold)}\) |
| Support | \(x \in (\gamma, +\infty)\) |
| Optimization parameter | \(\gamma = \text{Offset (Threshold)}\) |
| Probability density function (pdf) | \(f(x) = \dfrac{e^{\left(\frac{\ln(x-\gamma)-\mu}{\sigma}\right)}}{\sigma(x-\gamma)\left[1+e^{\left(\frac{\ln(x-\gamma)-\mu}{\sigma}\right)}\right]^2}\) |
Note
The 2-parameter LogLogistic is equivalent to the 3-parameter LogLogistic when offset (threshold) is zero. Much like the Weibull distribution, care must be used with the parameters of the LogLogistic. Different text and software will reverse the terms shape and scale.