Chapter 15: 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, as discussed in Meeker and Escobar . For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.
Loglogistic Probability Density Function
The loglogistic distribution is a 2-parameter distribution with parameters and . The pdf for this distribution is given by:
where , and .
Mean, Median and Mode
The mean of the loglogistic distribution, , is given by:
Note that for does not exist.
The median of the loglogistic distribution, , is given by:
The mode of the loglogistic distribution, , if is given by:
The Standard Deviation
The standard deviation of the loglogistic distribution, , is given by:
Note that for the standard deviation does not exist.
The Loglogistic Reliability Function
The reliability for a mission of time , starting at age 0, for the loglogistic distribution is determined by:
The unreliability function is:
The loglogistic Reliable Life
The logistic reliable life is:
The loglogistic Failure Rate Function
The loglogistic failure rate is given by:
- decreases monotonically and is convex. Mode and mean do not exist.
- decreases monotonically and is convex. Mode and mean do not exist. As ,
- As ,
- The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
- The pdf starts at zero, increases to its mode, and decreases thereafter.
- As increases, while is kept the same, the pdf gets stretched out to the right and its height decreases, while maintaining its shape.
- As decreases,while is kept the same, the pdf gets pushed in towards the left and its height increases.
- increases till and decreases thereafter. is concave at first, then becomes convex.
The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Parameter Estimation.
Bounds on the Parameters
The lower and upper bounds , are estimated from:
For paramter , is treated as normally distributed, and the bounds are estimated from:
where is defined by:
If is the confidence level, then for the two-sided bounds, and for the one-sided bounds.
The variances and covariances of and are estimated as follows:
where is the log-likelihood function of the loglogistic distribution.
Bounds on Reliability
The reliability of the logistic distribution is:
Here , , , therefore and also is changing from till .
The bounds on are estimated from:
The upper and lower bounds on reliability are:
Bounds on Time
The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:
The upper and lower bounds are then found by:
Determine the loglogistic parameter estimates for the data given in the following table.
Set up the folio for times-to-failure data that includes interval and left censored data, then enter the data. The computed parameters for maximum likelihood are calculated to be:
For rank regression on :
For rank regression on :