Template:LoglogisticDistribution: Difference between revisions
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f(t)= & \frac{e^z}{\sigma{t}{(1+{e^z})^2}} \\ | f(t)= & \frac{e^z}{\sigma{t}{(1+{e^z})^2}} \\ | ||
z= & \frac{t'-{\mu }}{\sigma } \\ | z= & \frac{t'-{\mu }}{\sigma } \\ | ||
f(t)\ge & 0, | f(t)\ge & 0, t>0, {{\sigma}}>0, \\ | ||
{t}'= & ln(t) | {t}'= & ln(t) | ||
\end{align}</math> | \end{align}</math> | ||
<br> | <br> | ||
where, | |||
::<math>\begin{align} | ::<math>\begin{align} | ||
\mu= & \text{scale parameter} \\ | \mu= & \text{scale parameter} \\ | ||
Revision as of 23:03, 3 February 2012
The Loglogistic Distribution
As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.
The [math]\displaystyle{ pdf }[/math] of the loglogistic distribution is given by:
- [math]\displaystyle{ \begin{align} f(t)= & \frac{e^z}{\sigma{t}{(1+{e^z})^2}} \\ z= & \frac{t'-{\mu }}{\sigma } \\ f(t)\ge & 0, t\gt 0, {{\sigma}}\gt 0, \\ {t}'= & ln(t) \end{align} }[/math]
where,
- [math]\displaystyle{ \begin{align} \mu= & \text{scale parameter} \\ \sigma=& \text{shape parameter} \end{align} }[/math]
The loglogistic distribution and its characteristics are presented in more detail in Chapter 10.