# Three parameter generalized Pareto distribution and L-moments `GENPAR` provides the link between L-moments of a sample and the three parameter generalized Pareto distribution. ```r f.genpar (x, xi, alfa, k) F.genpar (x, xi, alfa, k) invF.genpar (F, xi, alfa, k) Lmom.genpar (xi, alfa, k) par.genpar (lambda1, lambda2, tau3) rand.genpar (numerosita, xi, alfa, k) ``` ## Arguments - `x`: vector of quantiles - `xi`: vector of genpar location parameters - `alfa`: vector of genpar scale parameters - `k`: vector of genpar shape parameters - `F`: vector of probabilities - `lambda1`: vector of sample means - `lambda2`: vector of L-variances - `tau3`: vector of L-CA (or L-skewness) - `numerosita`: numeric value indicating the length of the vector to be generated ## Details See [https://en.wikipedia.org/wiki/Pareto_distribution](https://en.wikipedia.org/wiki/Pareto_distribution) for an introduction to the Pareto distribution. Definition Parameters (3): $\xi$ (location), $\alpha$ (scale), $k$ (shape). Range of $x$: $\xi < x \le \xi + \alpha / k$ if $k>0$; $\xi \le x < \infty$ if $k \le 0$. Probability density function: $$ f(x) = \alpha^{-1} e^{-(1-k)y} $$ where $y = -k^{-1}\log\{1 - k(x - \xi)/\alpha\}$ if $k \ne 0$, $y = (x-\xi)/\alpha$ if $k=0$. Cumulative distribution function: $$ F(x) = 1-e^{-y} $$ Quantile function: $x(F) = \xi + \alpha[1-(1-F)^k]/k$ if $k \ne 0$, $x(F) = \xi - \alpha \log(1-F)$ if $k=0$. $k=0$ is the exponential distribution; $k=1$ is the uniform distribution on the interval $\xi < x \le \xi + \alpha$. L-moments L-moments are defined for $k>-1$. $$ \lambda_1 = \xi + \alpha/(1+k)] $$ $$ \lambda_2 = \alpha/[(1+k)(2+k)] $$ $$ \tau_3 = (1-k)/(3+k) $$ $$ \tau_4 = (1-k)(2-k)/[(3+k)(4+k)] $$ The relation between $\tau_3$ and $\tau_4$ is given by $$ \tau_4 = \frac{\tau_3 (1 + 5 \tau_3)}{5+\tau_3} $$ Parameters If $\xi$ is known, $k=(\lambda_1 - \xi)/\lambda_2 - 2$ and $\alpha=(1+k)(\lambda_1 - \xi)$; if $\xi$ is unknown, $k=(1 - 3 \tau_3)/(1 + \tau_3)$, $\alpha=(1+k)(2+k)\lambda_2$ and $\xi=\lambda_1 - (2+k)\lambda_2$. `Lmom.genpar` and `par.genpar` accept input as vectors of equal length. In `f.genpar`, `F.genpar`, `invF.genpar` and `rand.genpar` parameters (`xi`, `alfa`, `k`) must be atomic. ## Returns `f.genpar` gives the density $f$, `F.genpar` gives the distribution function $F$, `invF.genpar` gives the quantile function $x$, `Lmom.genpar` gives the L-moments ($\lambda_1$, $\lambda_2$, $\tau_3$, $\tau_4$), `par.genpar` gives the parameters (`xi`, `alfa`, `k`), and `rand.genpar` generates random deviates. ## Note For information on the package and the Author, and for all the references, see `nsRFA`. ## See Also `rnorm`, `runif`, `EXP`, `GENLOGIS`, `GEV`, `GUMBEL`, `KAPPA`, `LOGNORM`, `P3`; `DISTPLOTS`, `GOFmontecarlo`, `Lmoments`. ## Examples ```r data(hydroSIMN) annualflows summary(annualflows) x <- annualflows["dato"][,] fac <- factor(annualflows["cod"][,]) split(x,fac) camp <- split(x,fac)$"45" ll <- Lmoments(camp) parameters <- par.genpar(ll[1],ll[2],ll[4]) f.genpar(1800,parameters$xi,parameters$alfa,parameters$k) F.genpar(1800,parameters$xi,parameters$alfa,parameters$k) invF.genpar(0.7161775,parameters$xi,parameters$alfa,parameters$k) Lmom.genpar(parameters$xi,parameters$alfa,parameters$k) rand.genpar(100,parameters$xi,parameters$alfa,parameters$k) Rll <- regionalLmoments(x,fac); Rll parameters <- par.genpar(Rll[1],Rll[2],Rll[4]) Lmom.genpar(parameters$xi,parameters$alfa,parameters$k) ```

GENPAR function