Wakeby distribution

The Wakeby distribution[1] is a five-parameter probability distribution defined by its quantile function,

${\displaystyle W(p)=\xi +{\frac {\alpha }{\beta }}(1-(1-p)^{\beta })-{\frac {\gamma }{\delta }}(1-(1-p)^{-\delta })}$,

and by its quantile density function,

${\displaystyle W'(p)=w(p)=\alpha (1-p)^{\beta -1}+\gamma (1-p)^{-\delta -1}}$,

where ${\displaystyle 0\leq p\leq 1}$, ξ is a location parameter, α and γ are scale parameters and β and δ are shape parameters.

The Wakeby distribution has been used for modelling flood flows[2][3] and distribution of citation counts.[4] 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:

• ${\displaystyle \beta +\delta \geq 0}$
• Either ${\displaystyle \beta +\delta >0}$ or ${\displaystyle \beta =\gamma =\delta =0}$
• If ${\displaystyle \gamma >0}$, then ${\displaystyle \delta >0}$
• ${\displaystyle \gamma \geq 0}$
• ${\displaystyle \alpha +\gamma \geq 0}$

The domain of the Wakeby distribution is

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

With two 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):

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

where F is the cumulative distribution function and

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

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

An alternative to the above method is to define the PDF parametrically as ${\displaystyle (W(p),1/w(p)),\ 0\leq p\leq 1}$. This can be setup as a probability density function, ${\displaystyle f(x)}$, by solving for the unique ${\displaystyle p}$ in the equation ${\displaystyle W(p)=x}$ and returning ${\displaystyle 1/w(p)}$.