8. Assessing Product Reliability
8.2. Assumptions/Prerequisites
8.2.1. How do you choose an appropriate life distribution model?

8.2.1.2.

Extreme value argument

If component or system failure occurs when the first of many competing failure processes reaches a critical point, then Extreme Value Theory suggests that the Weibull Distribution will be a good model

It is well known that the Central Limit Theorem suggests that normal distributions will successfully model most engineering data when the observed measurements arise from the sum of many small random sources (such as measurement errors).  Practical experience validates this theory - the normal distribution "works" for many engineering data sets. 

Less known is the fact that Extreme Value Theory suggests that the Weibull distribution will successfully model failure times for mechanisms for which many competing similar failure processes are "racing" to failure and the first to reach it (i.e., the minimum of a large collection of roughly comparable random failure times) produces the observed failure time. Analogously, when a large number of roughly equivalent runners are competing and the winning time is recorded for many similar races, these times are likely to follow a Weibull distribution. 

Note that this does not mean that anytime there are several failure mechanisms competing to cause a component or system to fail, the Weibull model applies. One or a few of these mechanisms may dominate the others and cause almost all of the failures. Then the "minimum of a large number of roughly comparable" random failure times does not apply and the proper model should be derived from the distribution models for the few dominating mechanisms using the competing risk model.

On the other hand, there are many cases in which failure occurs at the weakest link of a large number of similar degradation processes or defect flaws. One example of this occurs when modeling catastrophic failures of capacitors caused by dielectric material breakdown. Typical dielectric material has many "flaws" or microscopic sites where a breakdown will eventually take place. These sites may be thought of as competing with each other to reach failure first. The Weibull model, as extreme value theory would suggest, has been very successful as a life distribution model for this failure mechanism.