Issue 56, October 2005

Overview of the Gumbel, Logistic, Loglogistic and Gamma Distributions

Weibull++ introduces four more life distributions in addition to the Weibull-Bayesian 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 wear-out 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.

• As σ increases, the pdf spreads out and becomes shallower. As σ decreases, the pdf becomes taller and narrower.

• If times follow the Weibull distribution then the logarithms of times follow a Gumbel distribution. If t_i follows a Weibull distribution with β and η, then Ln(t_i) follows a Gumbel distribution with μ = ln (η) and σ = 1/β.

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 left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. 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:

• f(T) decreases monotonically and is convex.

For σ = 1:

• f(T) decreases monotonically and is convex.

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 time-to-first 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:

• Gamma becomes the exponential distribution.

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 e-Handbook 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.
 

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