cdfwak {lmom}

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.

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.

See Also

cdfgpa for the generalized Pareto distribution.

cdfexp for the exponential distribution.

Examples

# 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))