Parameterizations of Weibull Distribution

Shape and Scale

One of the most popular Reliability Engineering software package is Weibull++ (r) made by Reliasoft. The company also maintains the website [www.weibull.com]. It provides wealth of information of reliability engineering, and especially, Weibull distribution and its applications. In their implentation, the Weibull distribution has a characteristic life parameter \(\eta\) and a shape factor \(\beta\), and the hazard function is \[ h \left( t \right) = \frac{\beta} {\eta} \left( \frac{t}{\eta}\right)^\beta\]

• where \(\eta\) is the scale parameter, or characteristic life
• and \(\beta\) is the shape parameter (or slope) The reliability function is thus \[ R \left( t \right) = e^ {- \left( \frac{t}{\eta} \right) ^ \beta} \]

It is easy to see:

• when \(\beta = 1\), the h(t) is a flat line, a constant;
• when \(\beta > 1\), the h(t) is monotonously increasing over time;
• when \(\beta < 1\), the h(t) is monotonously increasing over time;
• While \(\beta\) determines the shaple of curve, the \(\eta\) determines the scale.
• \(\eta\) is the time that no matter what \(\beta\) is, \(R \left( t \right) = e^{-1} = 63\%\), thus the name charasteristic life.

Through my graduate schools, we used same parameterization in the textbooks ( Nelson and Modarres), except that \(\alpha\) took the place of \(\eta\). Give it a few years, I adjusted.

You can hear debate about \(\beta\) and \(\eta\) of a component around my office. Engineers love this simplicity.

Until there is none.

Most Popular Stastical Software

There is no denial that the by far most widely used statistical software is Excel, even though a unreliable one. Let’s look at the Weibull distribution implemented in Excel:

WEIBULL(x,alpha,beta,cumulative)

The WEIBULL function syntax has the following arguments:

X Required. The value at which to evaluate the function.

Alpha, Beta Required, parameters to the distribution.

Cumulative Required. Determines the form of the function, if True, returns CDP, if false, pdf..

The equation for the Weibull cumulative distribution function is: \[ F\left( t; \alpha, \beta \right) = 1 - e^ {- \left( \frac{t}{\beta} \right) ^ \alpha}\]

Well, it not too bad, just my world upside down. \(\alpha\) is \(\beta\), and \(\beta\) is \(\alpha\). In excel, \(\beta\) is scale, and \(\alpha\) is shape.

Also, important to note that in Excel function call, only order matters, and you cannot force call by argument as in R.

The Upstart

Welcome to R

The Weibull distribution with shape parameter a and scale parameter b has density given by
f(x) = (a/b) (x/b)^(a-1) exp(- (x/b)^a)
for x > 0. The cumulative distribution function is F(x) = 1 - exp(- (x/b)^a) on x > 0

So, a is shape, or \(\alpha\). And b is scale, or \(\beta\), as in Excel, not Nelson. When calling function with argument, as most do in R, it is less confusing.

Shape = 1 means constant as in most implementations, and scale is the characteristic life.

t <- 0:1000
plot(t, pweibull(t, shape = 1, scale = 500))

And python See the doc from numpy:

The probability density for the Weibull distribution is \[p\left(x\right)=\frac{a}{\lambda}\cdot \left(\frac{x}{\lambda}\right)^{a-1}\cdot e^{-\left(x/b\right)^a}\] where \(a\) is the shape and \(\lambda\) the scale.

Same equation as r, except \(\lambda\) is in the place of \(b\).