In addition to the distributions mentioned, the following additional distributions, even though not as frequently used in Life Data Analysis, have a variety of applications and can be found in many statistical references. They are included in Weibull++ as well as discussed in this reference.
The mixed Weibull distribution is commonly used for modeling the behavior of components or systems exhibiting multiple failure modes (mixed populations). It gives the global picture of the life of a product by mixing different Weibull distributions for different stages of the product's life and is defined by:
where the value of S is equal to the number of subpopulations. Note that this results in a total of (3 • S - 1) parameters. In other words, each population has a pi, portion or mixing weight for the ith population, a βi, or shape parameter for the ith population and an ni, or scale parameter for the ith population. Note that the parameters are reduced to (3 • S - 1), given the fact that the following condition can also be used:
The mixed Weibull distribution and its characteristics are presented in more detail in the Other Distributions chapter.
While not as frequently used for modeling life data as the distributions discussed previously, the generalized gamma distribution does have the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution's parameters and also offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters μ, σ and λ. The pdf of the distribution is given by,
where Γ(x) is the gamma function, defined by:
This distribution behaves as do other distributions based on the values of the parameters. For example, if λ = 1, the distribution is identical to the Weibull distribution. If both λ = 1 and σ = 1, the distribution is identical to the exponential distribution and for λ = 0, it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, gamma distribution has applications in Bayesian analysis as a prior distribution and is also commonly used in queuing theory.
The pdf of the gamma distribution is given by:
(11)
where:
μ
= scale parameter
k = shape parameter
where 0 < t < 
, - <
μ
< and k > 0.
The gamma distribution and its characteristics are presented in more detail in the Other Distributions chapter.
The logistic distribution has a shape very similar to the normal distribution (i.e. bell shaped), but with heavier tails. Since the logistic distribution has closed form solutions for the reliability, cdf and failure rate functions, it is sometimes preferred over the normal distribution, where these functions can only be obtained numerically.
The pdf of the logistic distribution is given by:
(12)
where:
μ
= location parameter (also denoted as 
)
σ = scale parameter
The logistic distribution and its characteristics are presented in more detail in the Other Distributions chapter.
As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.
The pdf of the loglogistic distribution is given by:
(13)
where,
μ'
= scale parameter
σT' = shape parameter
The loglogistic distribution and its characteristics are presented in more detail in the Other Distributions chapter.
The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type 1) distribution. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (few weak units fail under low stress, while the rest fail at higher stresses). 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 pdf of the Gumbel distribution is given by:
(14)
where,
μ
= location parameter
σ = scale parameter
The Gumbel distribution and its characteristics are presented in more detail in the Other Distributions chapter.
See Also:
Statistical Background
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