Two-Parameter Weibull

The required parameters for the two-parameter Weibull distribution are the slope and characteristic life. This Weibull distribution provides reasonably accurate failure analysis and failure forecasts with extremely small samples. It has the special capability to diagnose failure types, such as infant mortality (particularly for electronics), age-independent failures (accidents and natural occurrences) or wear-out type mechanisms (bearings, filters, etc.). The two-parameter Weibull distribution is recommended if the failure rate decreases (burn-in period) or increases (wear-out period) over time, or if the failure rate remains constant (random failure period).

Exponential

The only parameter required for the exponential distribution is the failure rate. The exponential distribution can be viewed as a special case of the Weibull distribution, where the β value is known to equal 1. When the failure rate for a component is constant, then its reliability is best described by the Weibull or exponential distribution. A constant failure rate leads to the memoryless property, which states that the remaining life of a used component is independent of its current age, thereby declaring that a used component as good as a new component. (The Weibull distribution is memoryless only when the β value equals 1.) Because the exponential distribution assumes that there is no infant mortality or wear-out period, the field data must be carefully tested to ensure that such assumptions are valid. For the exponential distribution, the MTTF is the reciprocal of the failure rate.

Rayleigh

The only parameter required for the Rayleigh distribution is the characteristic life. The Rayleigh distribution can be viewed as a special case of the Weibull distribution, where the β value is known to equal 2. It is, however, an important distribution in its own right, finding application not only in reliability problems but also in noise problems associated with communication systems. A single-parameter distribution similar to the exponential distribution, the Rayleigh distribution can be used to describe the root-mean-square (RMS) value of error sources. The Rayleigh distribution is recommended if the failure rate increases linearly with time.

WeiBayes

The only parameter required for the WeiBayes distribution is characteristic life. Also known as the one-parameter Weibull distribution, WeiBayes is a special case of the Weibull distribution where the βslope parameter (β) is defined based on prior knowledge. Related to Bayesian assumption, the WeiBayes distribution is a powerful method developed to solve the problems that occur when traditional Weibull analysis has large uncertainties. The WeiBayes distribution is more accurate than two-parameter Weibull distributions when the sample has fewer than 10 failures, and it is the only distribution that can be used when there are 0 failures. For example, after a design change corrects an existing failure mode, success data from tests can be used to determine a lower confidence bound for the Weibull line for the new design called a WeiBayes line. When parts exceed their design life, a Weibull analysis with no failures can be constructed to extend their life. Because the WeiBayes distribution can be used without the requirement of testing to failure, it is of extreme importance in situations where failures involve safety or extreme costs.

Three-Parameter Weibull

In addition to slope and characteristic life parameters, the three-parameter Weibull distribution requires a location parameter, t-zero (t0), that defines the location of the distribution in time. This third parameter provides for shifting the origin of the age scale and is only used if earlier two-parameter Weibull analysis has shown that it is appropriate. (For additional information, refer to Curved Data on Weibull Probability Plots.) When using the location parameter, the t0 value is either subtracted from or added to each age value prior to generating the Weibull probability plot. For example, if the probability of failure is zero for some given period of time, the origin of the age scale should be shifted from zero to time t0 to reflect this guaranteed failure-free period. The correction,t0, would be a positive value equal to the minimum time necessary for a failure to occur. To provide for some loss of life (reliability) before service officially begins, t0 can be a negative value. Negative corrections are helpful for situations where spare parts deteriorate while in storage. Rubber parts, chemicals and ball bearings, for example, all deteriorate with prolonged storage. When the t0 value applied to the data is correct, the resulting plot follows a straight line. Without prior experience, at least 20 failures are usually needed to do a distribution analysis using the three-parameter Weibull distribution.

Gumbel

In the 1920s, E. J. Gumbel was the first to seriously investigate extreme values in failure data, finding that there are only six separate extreme value distributions. His Type III smallest extreme value distribution is the same as the Weibull distribution. The Gumbel- (lower) distribution, which is also known as a Type I lower extreme value distribution, is an extreme minimum value distribution. The Gumbel+ (upper) distribution, which is also known as a Type I upper extreme value distribution, is an extreme maximum value distribution. Gumbel distributions are recommended when failure data is a result of rare events and failure values are extreme. Examples include natural disasters and maximum guest loads. Because Gumbel distributions (and normal distributions) can predict negative life for high reliability requirements, an impossibility with life data, care must be taken when using them to model life data.