Two special cases of the Weibull model arise from the physics of certain processes. The Exponential distribution has a Weibull shape parameter, = 1, and = 2, produces the Rayleigh distribution.
The smallest extreme value (SEV) and largest extreme value (LEV) are also related to the Weibull distribution.
The Weibull model describes the fraction failing, F(X), before some time X, which depends on a location parameter, >0 and a shape (and scale) parameter, >0. Unlike the lognormal distribution whose shape remains unchanged as its scale parameter changes, changing the scale of the Weibull model unavoidably also changes its shape as well.
The SEV is to the Weibull, as the normal is to the logmormal, i.e., the SEV is logWeibull. That is, if X has a Weibull distribution, then log(X) has an SEV distribution. In fact, you can view the SEV is simply a reparameterization of the Weibull.
Look at the figure above. The scale on the bottom is logarithmic. The scale on the top is Cartesian. The data are plotted as log(X) but can be read across the bottom as X.
It is convenient to write location, scale models as
In its familiar form Weibull isn't a location, scale model, but the SEV is, i.e.
A little algebraic manipulation will show that
The distribution of the largest extreme value, not surprisingly, has a multiplicative inverse relationship with the smallest extreme value: if log(X) is SEV, then log(1/X ) = -log(X) is LEV.
Viewed differently, if Y = log(X) has a largest extreme value distribution, LEV(), then -Y = SEV(-)
(... more to come)