Wakeby distribution

Description

Distribution function and quantile function of the Wakeby distribution.

Usage

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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 {1-(1-F)^beta}/beta - gamma {1-(1-F)^(-delta)}/delta .

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 >= 0;

  • alpha + gamma >= 0.

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

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# 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))