Weibull Distribution
The Weibull distribution (Weibull, 1939, 1951; see also Lieblein, 1955)
has density function (for positive parameters b, c, and q):
f(x) = c/b*[(x-q)/b](c-1)
* e{-[(x-q)/b]^c}, for q < x, b > 0, c
> 0
where
|
b |
is the
scale parameter of the distribution |
|
c |
is the
shape parameter of the distribution |
|
q |
is the
location parameter of the distribution |
|
e |
is the base of the natural logarithm, sometimes called Euler's
e (2.71...) |
Note that in Survival
Analysis, instead of the scale parameter b, the inverse 1/b
= Lambda is often estimated.
Also, if you use the Life Table Analysis facilities to estimate the parameters
of the Weibull distribution (using weighted least squares methods), the
program will estimate and report the parameter L' = Lc
(Lambda to the power of c). Therefore,
when comparing the results computed by the Survival Analysis
module with those computed by, for example the Process Analysis
module, the estimates for the scale parameter will not be directly compatible.
The animation above shows the Weibull distribution as the shape parameter
increases (.5, 1, 2, 3, 4, 5, and 10).
For a complete listing of all distribution
functions, see Distributions
and Their Functions.