Example: The Effect of Beta on the Weibull pdf

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This example appears in the The Life Data Analysis Reference book.

The Effect of Beta on the Weibull pdf

The following figure shows the effect of different values of the shape parameter, β, on the shape of the [math]\displaystyle{ pdf }[/math]. As you can see, the shape can take on a variety of forms based on the value of β.

For [math]\displaystyle{ 0\lt \beta \leq 1 }[/math]:

• As t→0 (or γ), f(t)→∞.
• As t→∞, f(t)→0.
• f(t) decreases monotonically and is convex as it increases beyond the value of γ.
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• f(t) = 0 at (or γ).
• f(t) increases as [math]\displaystyle{ t\rightarrow \tilde{T} }[/math] (the mode) and decreases thereafter.
• For β < 2.6 the Weibull [math]\displaystyle{ pdf }[/math] is positively skewed (has a right tail), for 2.6 < β < 3.7 its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal [math]\displaystyle{ pdf }[/math] , and for β > 3.7 it is negatively skewed (left tail). The way the value of β relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for β = 0.999, f(0) = ∞, but for β = 1.001, f(0) = 0. This abrupt shift is what complicates MLE estimation when β is close to 1.