lmomkap {lmomco}

L-moments of the Kappa Distribution

Description

This function estimates the L-moments of the Kappa distribution given the parameters (\xi, \alpha, \kappa, and h) from parkap. The L-moments in terms of the parameters are complicated and are solved numerically. If the parameter k = 0 (is small or near zero) then let

d_r = \gamma + \log(-h) + \mathrm{digamma}(-r/h)\ \mbox{for}\ h < 0

d_r = \gamma + \log(r)\ \mbox{for}\ h = 0\ \mbox{(is small)}

d_r = \gamma + \log(h) + \mathrm{digamma}(1+r/h)\ \mbox{for}\ h > 0

or if k > -1 (nonzero) then let

g_r = \frac{\Gamma(1+k)\Gamma(-r/h-k)}{-h^k\,\Gamma(-r/h)}\ \mbox{for}\ h < 0

g_r = \frac{\Gamma(1+k)}{r^k} \times (1-0.5hk(1+k)/r)\ \mbox{for}\ h = 0\ \mbox{(is small)}

g_r = \frac{\Gamma(1+k)\Gamma(1+r/h)}{h^g\,\Gamma(1+k+r/h)}\ \mbox{for}\ h > 0

where r is L-moment order, \gamma is Euler's constant, and for h = 0 the term to the right of the multiplication is not in Hosking (1994) or Hosking and Wallis (1997) for exists within Hosking's FORTRAN code base.

The probability-weighted moments (\beta_r; pwm2lmom) for k = 0 (is small or near zero) are

r\beta_{r-1} = \xi + (\alpha/\kappa)[1 - d_r]

or if k > -1 (nonzero) then

r\beta_{r-1} = \xi + (\alpha/\kappa)[1 - g_r]

Usage

lmomkap(para, nmom=5)

Arguments

para

The parameters of the distribution.

nmom

The number of moments to compute. Default is 5.

Value

An R list is returned.

lambdas

Vector of the L-moments. First element is \lambda_1, second element is \lambda_2, and so on.

ratios

Vector of the L-moment ratios. Second element is \tau, third element is \tau_3 and so on.

trim

Level of symmetrical trimming used in the computation, which is 0.

leftrim

Level of left-tail trimming used in the computation, which is NULL.

rightrim

Level of right-tail trimming used in the computation, which is NULL.

source

An attribute identifying the computational source of the L-moments: “lmomkap”.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1994, The four-parameter kappa distribution: IBM Journal of Reserach and Development, v. 38, no. 3, pp. 251–258.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

parkap, cdfkap, pdfkap, quakap

Examples

lmr <- lmoms(c(123, 34, 4,78, 45, 234, 65, 2, 3, 5, 76, 7, 80))
lmomkap(parkap(lmr))