Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
• The Gumbel distribution is unbounded and can have its mass either in the lower
or upper portion of the distribution.
• The Fréchet distribution, in contrast, has most of its mass at low values, has a
lower limit, and has an unlimited upper tail.
• The Weibull distribution has most of its mass in the upper tail of the distribution
and takes on a maximum value.
The Generalized EV probability density function formulation is presented in Equation 1.
It has three parameters: the location (µ), scale (σ) and shape (ξ) parameters. These
parameters can be used to distinguish the three distribution types. Similar to the mean and
standard deviation of a normal distribution, the location parameter µdetermines the
"location" of the distribution, shifting the distribution to the right or left, without
changing the shape of the distribution. The scale parameterσdetermines the spread of the
distribution. The shape parameter ξ controls the shape of the distribution—in particular
the tail behavior of the distribution.
The Figure C.1 below illustrates the standard shapes of the respective distributions for
µ=0, σ=1 and ξ=-0.5 (Weibull), ξ= 0.5 (Fréchet) and ξ→0 (Gumbel). Since these are the
standard distributions, the distributions are located over zero and have a standard scale
parameter of one, with the lower bound for the Fréchet distribution at -2 and the upper
bound for the Weibull distribution at +2. For rare events, the Fréchet or Gumbel
distributions have the most appropriate shapes since most of the mass will occur at the
lower tail of the distribution and the extreme observations will occur in the upper tail. In
practice, the Gumbel distribution is the distribution that is most widely used to
characterize rare events because it can be estimated using a two parameter specification
(location and scale). The Gumbel probability density function is presented in Equation 2.
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-3