Wakeby distribution

Distribution function and quantile function of the Wakeby distribution.

cdfwak(x, para = c(0, 1, 0, 0, 0))
quawak(f, para = c(0, 1, 0, 0, 0))

Arguments

x: Vector of quantiles.
f: Vector of probabilities.
para: Numeric vector containing the parameters of the distribution, in the order $\xi, \alpha, \beta, \gamma, \delta$.

Details

The Wakeby distribution with parameters $\xi$, $\alpha$, $\beta$, $\gamma$ and $\delta$ has quantile function $$x(F)=\xi+{\alpha\over\beta}\lbrace1-(1-F)^\beta\rbrace-{\gamma\over\delta}\lbrace1-(1-F)^{-\delta}\rbrace.$$

The parameters are restricted as in Hosking and Wallis (1997, Appendix A.11):

either $\beta+\delta>0$ or $\beta=\gamma=\delta=0$;
if $\alpha=0$ then $\beta=0$;
if $\gamma=0$ then $\delta=0$;
$\gamma\ge0$;
$\alpha+\gamma\ge0$.

The distribution has a lower bound at $\xi$ and, if $\delta<0$, an upper bound at $\xi+\alpha/\beta-\gamma/\delta$.

The generalized Pareto distribution is the special case $\alpha=0$ or $\gamma=0$. The exponential distribution is the special case $\beta=\gamma=\delta=0$. The uniform distribution is the special case $\beta=1$, $\gamma=\delta=0$.

Value

cdfwak gives the distribution function; quawak gives the quantile function.

Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

References

Hosking, J. R. M. and Wallis, J. R. (1997). Regional frequency analysis: an approach based on L-moments, Cambridge University Press, Appendix A.11.

Examples