cdfgpa.RdDistribution function and quantile function of the generalized Pareto distribution.
The generalized Pareto distribution with location parameter \(\xi\), scale parameter \(\alpha\) and shape parameter \(k\) has distribution function $$F(x)=1-\exp(-y)$$ where $$y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace,$$ with \(x\) bounded by \(\xi+\alpha/k\) from below if \(k<0\) and from above if \(k>0\), and quantile function $$x(F)=\xi+{\alpha\over k}\lbrace 1-(1-F)^k\rbrace.$$
The exponential distribution is the special case \(k=0\). The uniform distribution is the special case \(k=1\).
cdfgpa gives the distribution function;
quagpa gives the quantile function.
The functions expect the distribution parameters in a vector,
rather than as separate arguments as in the standard R
distribution functions pnorm, qnorm, etc.
Two parametrizations of the generalized Pareto distribution are in common use. When Jenkinson (1955) introduced the generalized extreme-value distribution he wrote the distribution function in the form $$F(x) = \exp [ - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}].$$ Hosking and Wallis (1987) wrote the distribution function of the generalized Pareto distribution analogously as $$F(x) = 1 - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}$$ and that is the form used in R package lmom. A slight inconvenience with it is that the skewness of the distribution is a decreasing function of the shape parameter \(k\). Perhaps for this reason, authors of some other R packages prefer a form in which the sign of the shape parameter \(k\) is changed and the parameters are renamed: $$F(x) = 1 - \lbrace 1 + \xi ( x - \mu ) / \sigma) \rbrace ^{-1/\xi}.$$ Users should be able to mix functions from packages that use either form; just be aware that the sign of the shape parameter will need to be changed when converting from one form to the other (and that \(\xi\) is a location parameter in one form and a shape parameter in the other).
Hosking, J. R. M., and Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29, 339-349.
Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) of meteorological elements. Quarterly Journal of the Royal Meteorological Society, 81, 158-171.
# Random sample from the generalized Pareto distribution
# with parameters xi=0, alpha=1, k=-0.5.
quagpa(runif(100), c(0,1,-0.5))
#> [1] 1.219137266 0.676197071 0.417537998 0.109725489 0.345077168
#> [6] 1.398625971 5.446845392 0.027168402 1.013371663 3.237354803
#> [11] 0.080494287 0.244258838 0.816872482 0.327875376 0.617033240
#> [16] 0.670586402 0.211839909 0.083252259 0.018010347 1.836094262
#> [21] 9.713148720 0.933628749 0.190659042 0.060843622 0.719488145
#> [26] 1.256442958 1.141995564 1.945289202 0.899708569 0.819490113
#> [31] 0.014883304 7.749618278 0.704405184 12.716966077 0.964037094
#> [36] 0.500082819 0.837367867 0.079089285 9.507161840 1.373402346
#> [41] 14.098419632 0.936522800 2.091523268 1.030253386 2.249113545
#> [46] 1.328994544 0.537876395 0.499083845 3.067456858 0.713499843
#> [51] 0.207331123 8.880046003 1.735949165 0.025360364 1.272347380
#> [56] 2.821325639 1.347633310 1.704238937 3.562580894 0.359995090
#> [61] 0.199562395 0.119602859 1.109183814 3.176439151 5.139496560
#> [66] 1.511810014 0.481817801 0.008714705 0.651155748 1.813400383
#> [71] 0.608771471 0.984284934 0.177466895 0.044439768 1.561471918
#> [76] 0.870065269 6.581861308 3.049103717 0.068221465 2.100555906
#> [81] 0.083315534 0.148560724 2.584650846 1.013002479 1.875864811
#> [86] 0.480113795 13.191966977 0.601252879 0.501708182 1.392804728
#> [91] 0.161132169 1.539953949 3.331644415 2.978730085 3.805348314
#> [96] 0.290690376 1.081049143 4.259252246 0.470131869 2.089801933