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).
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:
To obtain the location parameter, γ :
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.
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++):
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