Wakeby distribution

The Wakeby distribution is a five-parameter probability distribution defined by its quantile function,
and by its quantile density function,
where, ξ is a location parameter, α and γ are scale parameters and
β and δ are shape parameters.
This distribution was first proposed by John C. Houghton while under the supervision of Harold A. Thomas Jr., who named it after Wakeby Pond in Cape Cod.

Applications

The Wakeby distribution has been used for modeling distributions of

flood flows,
citation counts,
extreme rainfall,
tidal current speeds,
peak flows of rivers,
and extreme wind speeds.
Parameters and domain

The following restrictions apply to the parameters of this distribution:

Either or
If, then

The domain of the Wakeby distribution is

to, if and
to, if or

With two shape parameters, the Wakeby distribution can model a wide variety of shapes.

CDF and PDF

The cumulative distribution function is computed by numerically inverting the quantile function given above. The probability density function is then found by using the following relation :
where F is the cumulative distribution function and
An implementation that computes the probability density function of the Wakeby distribution is included in the Dataplot scientific computation library, as routine WAKPDF.
An alternative to the above method is to define the PDF parametrically as. This can be set up as a probability density function,, by solving for the unique in the equation and returning.