Template:LoglogisticDistribution: Difference between revisions
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===The Loglogistic Distribution=== | === 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. | |||
<br> | 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. <br>The <span class="texhtml">''pdf''</span> of the loglogistic distribution is given by: <br> | ||
The < | |||
<br> | |||
::<math> \begin{align} | ::<math> \begin{align} | ||
f(t)= & \frac{e^z}{\sigma{t}{(1+{e^z})^2}} \\ | f(t)= & \frac{e^z}{\sigma{t}{(1+{e^z})^2}} \\ | ||
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{t}'= & ln(t) | {t}'= & ln(t) | ||
\end{align}</math> | \end{align}</math> | ||
<br> | |||
where, | <br>where, | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
\mu= & \text{scale parameter} \\ | \mu= & \text{scale parameter} \\ | ||
\sigma=& \text{shape parameter} | \sigma=& \text{shape parameter} | ||
\end{align}</math> | \end{align}</math> | ||
<br> | |||
The loglogistic distribution and its characteristics are presented in | <br>The loglogistic distribution and its characteristics are presented in detail in the chapter [[The Loglogistic]]. <br> | ||
<br> | |||
Revision as of 16:17, 12 March 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 pdf 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 detail in the chapter The Loglogistic.