Template:The effect of beta on the Weibull pdf: Difference between revisions
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![The effect of the Weibull shape parameter on the [math]\displaystyle{ pdf }[/math].](/images/thumb/b/bb/WB.8_weibull_pdf.png/400px-WB.8_weibull_pdf.png)
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[[Example: The Effect of Beta on the Weibull pdf]] | [[Example: The Effect of Beta on the Weibull pdf]] | ||
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:*<span class="texhtml">''f''(''t'')</span> increases as <math> t\rightarrow \tilde{T} </math> (the mode) and decreases thereafter. | :*<span class="texhtml">''f''(''t'')</span> increases as <math> t\rightarrow \tilde{T} </math> (the mode) and decreases thereafter. | ||
:*For <span class="texhtml">β < 2.6</span> the Weibull <math>pdf</math> is positively skewed (has a right tail), for <span class="texhtml">2.6 < β < 3.7</span> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal <math>pdf</math> , and for <span class="texhtml">β > 3.7</span> it is negatively skewed (left tail). The way the value of <span class="texhtml">β</span> 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 <span class="texhtml">β = 0.999</span>, <span class="texhtml">''f''(0) = ∞</span>, but for <span class="texhtml">β = 1.001</span>, <span class="texhtml">''f''(0) = 0.</span> This abrupt shift is what complicates MLE estimation when <span class="texhtml">β</span> is close to one. | :*For <span class="texhtml">β < 2.6</span> the Weibull <math>pdf</math> is positively skewed (has a right tail), for <span class="texhtml">2.6 < β < 3.7</span> its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal <math>pdf</math> , and for <span class="texhtml">β > 3.7</span> it is negatively skewed (left tail). The way the value of <span class="texhtml">β</span> 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 <span class="texhtml">β = 0.999</span>, <span class="texhtml">''f''(0) = ∞</span>, but for <span class="texhtml">β = 1.001</span>, <span class="texhtml">''f''(0) = 0.</span> This abrupt shift is what complicates MLE estimation when <span class="texhtml">β</span> is close to one. | ||
Revision as of 21:50, 2 April 2012
Example: 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]. One can see that the shape of the can take on a variety of forms based on the value of β.
![The effect of the Weibull shape parameter on the [math]\displaystyle{ pdf }[/math].](/index.php/File:WB.8_weibull_pdf.png)
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 increases beyond the value of γ.
- The mode is non-existent.
For β > 1:
- 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 one.