Kappa distribution

Distribution function and quantile function of the kappa distribution.

cdfkap(x, para = c(0, 1, 0, 0))
quakap(f, para = c(0, 1, 0, 0))

Arguments

x: Vector of quantiles.
f: Vector of probabilities.
para: Numeric vector containing the parameters of the distribution, in the order $\xi, \alpha, k, h$ (location, scale, shape, shape).

Details

The kappa distribution with location parameter $\xi$, scale parameter $\alpha$ and shape parameters $k$ and $h$ has quantile function $$x(F)=\xi+{\alpha\over k}\biggl\lbrace1-\biggl({1-F^h \over h}\biggr)^k\biggr\rbrace.$$

Its special cases include the generalized logistic ($h=-1$), generalized extreme-value ($h=0$), generalized Pareto ($h=1$), logistic ($k=0$, $h=-1$), Gumbel ($k=0$, $h=0$), exponential ($k=0$, $h=1$), and uniform ($k=1$, $h=1$) distributions.

Value

cdfkap gives the distribution function; quakap gives the quantile function.

References

Hosking, J. R. M. (1994). The four-parameter kappa distribution. IBM Journal of Research and Development, 38, 251-258.

Hosking, J. R. M., and Wallis, J. R. (1997). Regional frequency analysis: an approach based on L-moments, Cambridge University Press, Appendix A.10.

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.

Examples

# Random sample from the kappa distribution
# with parameters xi=0, alpha=1, k=-0.5, h=0.25.
quakap(runif(100), c(0,1,-0.5,0.25))
#>   [1] -0.060183750  3.193037901 -0.277196415  2.403198344  0.120144531
#>   [6] -0.146214389  3.122975262 -0.417890710  2.201118466  0.096028726
#>  [11]  8.520039275 -0.444048713  0.375065973  0.587762007  2.795166773
#>  [16] -0.677978253 -0.643885590  2.203276013 -0.256619342 -0.568296656
#>  [21]  3.287906717 -0.546097496  0.295680998 10.253014873  0.765666546
#>  [26]  1.833945665 -0.169033319  1.013828336 -0.331915314  0.335422988
#>  [31] -0.340175518 -0.444995494  0.125164642  6.563387390  3.528465722
#>  [36] -0.129731150  1.168791556 -0.436661324  0.005796371  0.994083037
#>  [41]  0.341892099  1.479203442  3.902273100 -0.478019453  1.029721576
#>  [46]  2.867453005  0.117045284  0.993421785  0.382867248 -0.342969145
#>  [51]  0.553772695 -0.387876576  1.131181279 -0.360978683  0.534845821
#>  [56] -0.523446651  0.517096378 -0.659898678  2.601203792 -0.070906922
#>  [61] -0.159480755  0.204716992  1.038007730  6.733412838  0.057045598
#>  [66]  1.014839241  2.613674124 -0.383722216  8.964254782 -0.523019226
#>  [71] -0.173456898  1.072010740  0.518326525 -0.151073644 11.757776228
#>  [76]  1.164880589 -0.513423989  1.030366452  3.560389115  5.473354577
#>  [81]  0.263300853  0.492404454  0.088061472 -0.155061640  7.767527775
#>  [86]  0.106197877 -0.264758804  1.607491751 -0.332538034 -0.204365566
#>  [91]  2.933991936  0.071109690 -0.572341981 -0.703120529 -0.445536760
#>  [96] -0.767129251  1.679708479  5.147007132 12.936360443  0.224295953