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.
This section includes the following subsections:
The loglogistic distribution is a two-parameter distribution with parameters μ and σ. The pdf for this distribution is given by:
where:
and:
where 0 < t < 
, - < μ < and
0 < σ
< .
The mean of the loglogistic distribution, 
, is given by:
Note that for σ
1,
does not exist.
The median of the loglogistic distribution, 
, is given by:
The mode of the loglogistic distribution, 
, if σ
< 1,
is given by:
The standard deviation of the loglogistic distribution, σT, is given by:
Note that for σ
0.5,
the standard deviation does not exist.
The reliability for a mission of time T, starting at age 0, for the loglogistic distribution is determined by:
where:
The unreliability function is:
The loglogistic reliable life is:
The loglogistic failure rate is given by:
For σ > 1:
f(T) decreases monotonically and is convex. Mode and mean do not exist.
For σ = 1:
f(T) decreases monotonically and is convex. Mode and mean do not exist.
As 
, 
As 
, 
For 0 < σ < 1:
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 μ increases, while σ is kept the same, the pdf gets stretched out to the right and its height decreases, while maintaining its shape.
As μ decreases, while σ is kept the same, the pdf gets pushed in towards the left and its height increases.
λ(T)
increases till 
and decreases thereafter. λ(T)
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 the Confidence Bounds chapter.
The lower and upper bounds on the mean, 
, are estimated from:
For the standard deviation, 
, ln() is treated as normally
distributed, and the bounds are estimated from:
where Kα is defined by:
If δ is the
confidence level, then 
for the two-sided bounds, and
α
= 1 - δ
for the one-sided bounds.
The variances and covariances of 
and 
are
estimated as follows:
where Λ is the log-likelihood function of the loglogistic distribution.
The reliability of the logistic distribution is:
where:
Here 0 < t < , - < μ
< ,
0 < σ
< ,
therefore 0 < ln(t)
< 
and z
also is changing from - till
+ .The
bounds on z are estimated
from:
where:
or:
The upper and lower bounds on reliability are:
(21)
(22)
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:
where:
or:
Let:
then:
(23)
(24)
where:
or:
The upper and lower bounds are then found by:
(25)
(26)
Determine the loglogistic parameter estimates for the data given in Table 10.3.
Table 10.3- Test data
|
Data point index |
Last Inspected |
State End time |
|
1 |
105 |
106 |
|
2 |
197 |
200 |
|
3 |
297 |
301 |
|
4 |
330 |
335 |
|
5 |
393 |
401 |
|
6 |
423 |
426 |
|
7 |
460 |
468 |
|
8 |
569 |
570 |
|
9 |
675 |
680 |
|
10 |
884 |
889 |
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:
For rank regression on X:
For rank regression on Y:
See Also:
Other Distributions
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