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| ==Distribution Characteristics==
| | #REDIRECT [[The_Loglogistic_Distribution]] |
| For <math>\sigma >1</math> :
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| :* <math>f(t)</math> decreases monotonically and is convex. Mode and mean do not exist.
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| For <math>\sigma =1</math> :
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| :* <math>f(t)</math> decreases monotonically and is convex. Mode and mean do not exist. As <math>t\to 0</math> , <math>f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>
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| :* As <math>t\to 0</math> , <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>
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| For <math>0<\sigma <1</math> :
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| :* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
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| :* The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.
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| :* As <math>\mu </math> increases, while <math>\sigma </math> is kept the same, the <math>pdf</math> gets stretched out to the right and its height decreases, while maintaining its shape.
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| :* As <math>\mu </math> decreases,while <math>\sigma </math> is kept the same, the .. gets pushed in towards the left and its height increases.
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| :* <math>\lambda (t)</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}</math> and decreases thereafter. <math>\lambda (t)</math> is concave at first, then becomes convex.
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| [[Image:WB.15 loglogistic pdf.png|center|250px| ]] | |