Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 15, May 2002

Reliability Basics

Location Parameter of the Weibull Distribution

In last month's issue of Reliability HotWire, we had a detailed look at the Weibull distribution, which included the characteristics of the estimated parameters, β and η. However, the location parameter, γ, was not discussed in much detail. In many cases, the location parameter may be assumed to be zero, but in this month's issue we will take a closer look at the the location parameter and how/when to use the 3-parameter Weibull distribution.

Characteristics of the Location Parameter, γ

As the name implies, the location parameter, γ, locates the distribution along the abscissa. Changing the value of γ has the effect of "sliding" the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

Effect of the location parameter on the Weibull pdf

  • When γ = 0 the distribution starts at time t = 0, or at the origin.

  • If γ > 0 then the distribution starts at the location γ to the right of the origin.

  • If γ < 0  then the distribution starts at the location γ to the left of the origin.

  • γ provides an estimate of the earliest time-to-failure of the units under test.

  • γ must be less than or equal to the first time-to-failure.

  • The life period 0 to +γ is a failure free operating period.

  • γ may assume all values and provides an estimate of the earliest time at which a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the data collection period for the analysis. For example, failures might have occurred during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.

  • γ has the same units as t, such as hours, miles, cycles, actuations, etc.

Probability Plotting for the Location Parameter, γ

The third parameter of the Weibull distribution, γ, is utilized when the data points do not fall on a straight line, but on a concave up or down curve. The value for γ can be estimated manually by considering the following points:

  • If the curve for median ranks (MR) vs. tj is concave down and the curve for MR vs. (tj - t1) is concave up then there exists a γ such that 0 < γ< t1, or γ has a positive value.

  • If the curve for median ranks (MR) vs. tj and the curve for MR vs. (tj - t1) are both concave up then there exists a negative γ that will straighten out the curve of MR vs. tj.

  • If neither one of the previous two points prevails, then either reject the Weibull pdf as one capable of representing the data or proceed with multiple population (Mixed Weibull) analysis. Additional information regarding the Mixed Weibull distribution can be found at

To obtain the location parameter, γ :

  • Subtract the same arbitrary value, γ , from all of the times-to-failure and re-plot the data.

  • If the initial curve is concave up then subtract a negative γ from each failure time.

  • If the initial curve is concave down then subtract a positive γ from each failure time.

  • Repeat this process until the data set plots on an acceptable straight line.

The location parameter, γ, is the subtracted (positive or negative) value that places the points in an acceptable straight line. It is important to point out that subtracting a negative γ is equivalent to adding it. In the probability plot shown below γ = 86.56. The trial and error process of estimating this value can be time-consuming. Fortunately, Weibull++ can calculate the value of γ automatically. 

Weibull probability plot with gamma from ReliaSoft's Weibull++ software

Therefore, to straighten the original data line, the value of γ must be subtracted from each of the points. Note that when adjusting for γ, the x-axis scale for the straight line becomes (t - γ).

When to Use the 3-Parameter Weibull Distribution

There are two main points to consider when you are deciding whether or not to use the 3-parameter Weibull distribution. First of all, does it make sense? Can you justify the existence of having a value of γ that is not equal to zero for the product/component that is being tested? Second, do the points follow a curve or a straight line when considering the 2-parameter Weibull distribution? If the points follow a curve and you can justify a value of γ that is not equal to zero, then you might want to consider using the 3-parameter Weibull distribution. If the points seem to follow more of a straight line but you would still like to use the 3-parameter Weibull distribution, then be aware of the following points (analysis assumed to be conducted in Weibull++):

  • If there does not seem to be any curvature, then adjusting the points to be straight really has no meaning.

  • If you force an adjustment (i.e., fitting 3-parameter Weibull), the math algorithm tries to find a γ that will adjust the already "adjusted" points and it can go either way, positive or negative. The solution will be a γ that yields the best correlation coefficient.

  • The value of γ will probably be small since the line is already "adjusted."


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