# cdfwak: Wakeby distribution In lmom: L-Moments

 cdfwak R Documentation

## 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.

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))

lmom documentation built on Aug. 29, 2023, 9:07 a.m.