GUMBEL                 package:nsRFA                 R Documentation

_T_w_o _p_a_r_a_m_e_t_e_r _G_u_m_b_e_l _d_i_s_t_r_i_b_u_t_i_o_n _a_n_d _L-_m_o_m_e_n_t_s

_D_e_s_c_r_i_p_t_i_o_n:

     'GUMBEL' provides the link between L-moments of a sample and the
     two parameter Gumbel distribution.

_U_s_a_g_e:

     f.gumb (x, xi, alfa)
     F.gumb (x, xi, alfa)
     invF.gumb (F, xi, alfa)
     Lmom.gumb (xi, alfa)
     par.gumb (lambda1, lambda2)
     rand.gumb (numerosita, xi, alfa)

_A_r_g_u_m_e_n_t_s:

       x: vector of quantiles

      xi: vector of gumb location parameters

    alfa: vector of gumb scale parameters

       F: vector of probabilities

 lambda1: vector of sample means

 lambda2: vector of L-variances

numerosita: numeric value indicating the length of the vector to be
          generated

_D_e_t_a_i_l_s:

     See <URL:
     http://en.wikipedia.org/wiki/Fisher-Tippett_distribution> for an
     introduction to the Gumbel distribution.

     *Definition*

     Parameters (2): xi (location), alpha (scale).

     Range of x: -infty < x < infty.

     Probability density function:

    f(x) = alpha^{-1} exp[-(x-xi)/alpha] exp{- exp[-(x-xi)/alpha]}


     Cumulative distribution function:

                   F(x) = exp[-exp(-(x-xi)/alpha)]


     Quantile function: x(F) = xi - alpha log(-log F).

     *L-moments*


                     lambda_1 = xi + alpha gamma


                        lambda_2 = alpha log 2


                   tau_3 = 0.1699 = log(9/8)/ log 2


            tau_4 = 0.1504 = (16 log 2 - 10 log 3)/ log 2


     Here gamma is Euler's constant, 0.5772...

     *Parameters*


                        alpha=lambda_2 / log 2


                     xi = lambda_1 - gamma alpha


     'Lmom.gumb' and 'par.gumb' accept input as vectors of equal
     length. In 'f.gumb', 'F.gumb', 'invF.gumb' and 'rand.gumb'
     parameters ('xi', 'alfa') must be atomic.

_V_a_l_u_e:

     'f.gumb' gives the density f, 'F.gumb' gives the distribution
     function F, 'invF.gumb' gives the quantile function x, 'Lmom.gumb'
     gives the L-moments (lambda_1, lambda_2, tau_3, tau_4)),
     'par.gumb' gives the parameters ('xi', 'alfa'), and 'rand.gumb'
     generates random deviates.

_N_o_t_e:

     For information on the package and the Author, and for all the
     references, see 'nsRFA'.

_S_e_e _A_l_s_o:

     'rnorm', 'runif', 'EXP', 'GENLOGIS', 'GENPAR', 'GEV', 'KAPPA',
     'LOGNORM', 'P3'; 'DISTPLOTS', 'GOFmontecarlo', 'Lmoments'.

_E_x_a_m_p_l_e_s:

     data(hydroSIMN)
     annualflows[1:10,]
     summary(annualflows)
     x <- annualflows["dato"][,]
     fac <- factor(annualflows["cod"][,])
     split(x,fac)

     camp <- split(x,fac)$"45"
     ll <- Lmoments(camp)
     parameters <- par.gumb(ll[1],ll[2])
     f.gumb(1800,parameters$xi,parameters$alfa)
     F.gumb(1800,parameters$xi,parameters$alfa)
     invF.gumb(0.7686843,parameters$xi,parameters$alfa)
     Lmom.gumb(parameters$xi,parameters$alfa)
     rand.gumb(100,parameters$xi,parameters$alfa)

     Rll <- regionalLmoments(x,fac); Rll
     parameters <- par.gumb(Rll[1],Rll[2])
     Lmom.gumb(parameters$xi,parameters$alfa)

