Overview of the Gumbel,
Logistic, Loglogistic and Gamma Distributions
Weibull++ introduces four more life distributions
in addition to the WeibullBayesian distribution discussed in the previous
issue of
HotWire. These are the Gumbel, logistic, loglogistic and Gamma
distributions. In this article, we present an overview of these
distributions and discuss some of their characteristics and applications.
1  The Gumbel Distribution
The Gumbel distribution is also referred to as the Smallest Extreme Value
(SEV) distribution or the Smallest Extreme Value (Type I) distribution. The
Gumbel distribution's
pdf is skewed to the left, unlike the Weibull distribution's pdf
which is skewed to the right. The Gumbel distribution is appropriate for
modeling strength, which is sometimes skewed to the left (few weak units in
the lower tail, most units in the upper tail of the strength population).
The Gumbel distribution could also be appropriate for modeling the life of
products that experience very quick wearout after reaching a certain age.
The distribution of logarithms of times can often be modeled with the Gumbel
distribution (in addition to the more commonly used lognormal distribution).
The
pdf of the Gumbel distribution is given by:
where:
and:
μ = the location
parameter
σ = the scale parameter
The
reliability for a mission of time T for the Gumbel distribution is
given by:
Characteristics of the Gumbel Distribution
Some of the specific characteristics of the Gumbel distribution are:

The
shape of the Gumbel distribution is skewed to the left. The Gumbel
pdf has no shape parameter. This means that the Gumbel pdf has
only one shape, which does not change.

The
Gumbel pdf has location parameter μ which is equal to the
mode but differs from median and mean. This is because the Gumbel
distribution is not symmetrical about its μ.

As
μ decreases, the pdf is shifted to the left. As μ
increases, the pdf is shifted to the right.
2  The Logistic Distribution
The
logistic distribution has been used for growth models and is used in a
certain type of regression known as the logistic regression. It has also
applications in modeling life data. The shape of the logistic distribution
and the normal distribution are very similar [1]. There are some who argue
that the logistic distribution is inappropriate for modeling lifetime data
because the lefthand limit of the distribution extends to negative
infinity. This could conceivably result in modeling negative
timestofailure. However, provided that the distribution in question has a
relatively high mean and a relatively small location parameter, the issue of
negative failure times should not present itself as a problem.
The
pdf of the logstic distribution is given by:
where:
and:
μ = the location
parameter
σ = the scale parameter
The
reliability for a mission of time T for the Gumbel distribution is
given by:
Characteristics of the Logistic Distribution

The logistic distribution has no shape parameter. This means that the
logistic pdf has only one shape, the bell shape, and this shape
does not change. The shape of the logistic distribution is very similar
to that of the normal distribution.

The mean, μ, or the mean life or the MTTF, is also the location
parameter of the logistic pdf , as it locates the pdf
along the abscissa.

As μ decreases, the pdf is shifted to the left.

As μ increases, the pdf is shifted to the right.

As σ decreases, the pdf gets pushed toward the mean, or it
becomes narrower and taller.

As σ increases, the pdf spreads out away from the mean, or
it becomes broader and shallower.

The main difference between the normal distribution and the logistic
distribution lies in the tails and in the behavior of the failure rate
function. The logistic distribution has slightly longer tails compared
to the normal distribution. Also, in the upper tail of the logistic
distribution, the failure rate function levels out for large t
approaching 1/σ.
3  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 [1]. For example, the
mathematical reasoning for the construction of the probability plotting
scales is very similar for these two distributions.
The
pdf of the logistic distribution is given by:
where:
and:
μ = the scale parameter
σ = the shape parameter
The
reliability for a mission of time T for the logistic distribution is
given by:
Characteristics of the Loglogistic Distribution
For
0 < σ < 1:

The shape of the loglogistic distribution is very similar to that of the
lognormal distribution and the Weibull distribution.

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.
For
σ > 1:
For
σ = 1:
4  The Gamma Distribution
The gamma distribution is a flexible life distribution model that may offer
a good fit to some sets of failure data. It is not, however, widely used as
a life distribution model for common failure mechanisms. The gamma
distribution does arise naturally as the timetofirst fail distribution for
a system with standby exponentially distributed backups, and is also a good
fit for the sum of independent exponential random variables. The gamma
distribution is sometimes called the Erlang distribution, which is used
frequently in queuing theory applications. [2]
The
pdf of the logistic distribution is given by:
where:
and:
e^{μ} = the scale
parameter
k = the shape parameter
The
reliability for a mission of time T for the logistic distribution is
given by:
Characteristics of the Gamma Distribution
For
0 < k < 1:

The pdf decreases monotonically and is convex.

As μ increases, the pdf gets stretched out to the right and
its height decreases while maintaining its shape.

As μ decreases, the pdf shifts towards the left and
its height increases.
For
k = 1:
For
k > 1:

If k ≤ 2 then the pdf has one inflection
point.

If k > 2 then the pdf has two inflection
points.

For a fixed k, as μ increases, the pdf starts to look
more like a straight angle.
References:
1. Meeker, W.Q., and Escobar, L.A., Statistical Methods for Reliability
Data, John Wiley &\Sons, Inc., New York, 1998.
2. NIST/SEMATECH eHandbook of
Statistical Methods,
http://www.itl.nist.gov/div898/handbook/, September, 2005.
Note:
Further details about calculating various reliability metrics, the
characteristics of the distribution, estimating parameters and confidence
bounds are available in the upcoming ReliaSoft Life Data Analysis
Reference book.
