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.