lmom (version 2.8)

cdfwak: Wakeby distribution

Description

Distribution function and quantile function of the Wakeby distribution.

Usage

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 \(\xi, \alpha, \beta, \gamma, \delta\).

Value

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

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 \(\beta=1\), \(\gamma=\delta=0\).

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.

See Also

cdfgpa for the generalized Pareto distribution.

cdfexp for the exponential distribution.

Examples

# NOT RUN {
# Random sample from the Wakeby distribution
# with parameters xi=0, alpha=30, beta=20, gamma=1, delta=0.3.
quawak(runif(100), c(0,30,20,1,0.3))
# }