Template:Loglogistic distribution

The Loglogistic Distribution
As may be indicated by the name, the loglogistic distribution has certain similarities to the logistic distribution. A random variable is loglogistically distributed if the logarithm of the random variable is logistically distributed. Because of this, there are many mathematical similarities between the two distributions [27]. For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.

A LogLogistic Distribution Example
Determine the loglogistic parameter estimates for the data given in Table 10.3.

$$\overset – {\mathop{\text{Table 10}\text{.3 - Test data}}}\,$$

$$\begin{matrix} \text{Data point index} & \text{Last Inspected} & \text{State End time} \\ \text{1} & \text{105} & \text{106} \\ \text{2} & \text{197} & \text{200} \\ \text{3} & \text{297} & \text{301} \\ \text{4} & \text{330} & \text{335} \\ \text{5} & \text{393} & \text{401} \\ \text{6} & \text{423} & \text{426} \\ \text{7} & \text{460} & \text{468} \\ \text{8} & \text{569} & \text{570} \\ \text{9} & \text{675} & \text{680} \\ \text{10} & \text{884} & \text{889} \\ \end{matrix}$$

Using Times-to-failure data under the Folio Data Type and the My data set contains interval and/or left censored data under Times-to-failure data options to enter the above data, the computed parameters for maximum likelihood are calculated to be:


 * $$\begin{align}

& {{{\hat{\mu }}}^{\prime }}= & 5.9772 \\ & {{{\hat{\sigma }}}_}= & 0.3256 \end{align}$$

For rank regression on $$X\ \ :$$


 * $$\begin{align}

& \hat{\mu }= & 5.9281 \\ & \hat{\sigma }= & 0.3821 \end{align}$$

For rank regression on $$Y\ \ :$$


 * $$\begin{align}

& \hat{\mu }= & 5.9772 \\ & \hat{\sigma }= & 0.3256 \end{align}$$