Wakeby distribution

The Wakeby distribution is a five-parameter probability distribution defined by the transformation

$X =\xi + \frac{\alpha}{\beta} (1 - (1-U)^{\beta}) - \frac{\gamma}{\delta} (1 - (1-U)^{-\delta})$

where U is a standard uniform random variable. That is, the above equation defines the quantile function for the Wakeby distribution.^[1] The parameters β, γ and δ are shape parameters. The parameters ξ and α are location parameters.

The Wakeby distribution has been used for modelling flood flows.^[2] This distribution was first proposed by Harold A. Thomas Jr., who named it after Wakeby Pond in Cape Cod.

The following restrictions apply to the parameters of this distribution:

• $\beta + \delta \ge 0$
• Either $\beta + \delta > 0$ or $\beta = \gamma = \delta = 0$
• If $\gamma > 0$, then $\delta > 0$
• $\gamma \ge 0$
• $\alpha + \gamma \ge 0$

The domain of the Wakeby distribution is

• $\xi$ to $\infty$, if $\delta \ge 0$ and $\gamma > 0$
• $\xi$ to $\xi + (\alpha/ \beta) - (\gamma/ \delta)$, if $\delta < 0$ or $\gamma = 0$

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

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 (given on page 46 of Johnson, Kotz, and Balakrishnan):

$f(x) = \frac{(1 - F(x))^{(\delta+1)}}{\alpha t + \gamma}$

where F is the cumulative distribution function and

$t = (1 - F(x))^{(\beta + \delta)}$

An implementation that computes the probability density function of the Wakeby distribution is included in the Dataplot scientific computation library, as routine WAKPDF.^[1]

See also

• Generalized Pareto distribution

References

1. "Dataplot reference manual: WAKPDF". NIST. Retrieved 20 August 2015.
2. John C. Houghton (October 14, 1977). "Birth of a Parent: The Wakeby Distribution for Modeling Flood Flows; Working Paper No. MIT-EL77-033WP" (PDF). MIT.

External links

• Discussion of the naming of the distribution on Stack Exchange

Note: this work is based on a NIST document that is in the public domain as a work of the U.S. federal government