gpdFamily             package:fExtremes             R Documentation

_G_P_D _D_i_s_t_r_i_b_u_t_i_o_n_s _f_o_r _E_x_t_r_e_m_e _V_a_l_u_e _T_h_e_o_r_y

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

     This is a collection of distribution functions used in extreme 
     value theory. The functions compute density, distribution
     function,  quantile function and generate random deviates for the
     Generalized  Pareto Distribution GPD. 

     The functions are:

       1  'dgpd'  Density of the GPD Distribution,
       2  'pgpd'  Probability function of the GPD Distribution,
       3  'qgpd'  Quantile function of the GPD Distribution,
       4  'rgpd'  Random variates from the GPD Distribution.

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

     dgpd(x, xi = 1, mu = 0, beta = 1) 
     pgpd(q, xi = 1, mu = 0, beta = 1) 
     qgpd(p, xi = 1, mu = 0, beta = 1) 
     rgpd(n, xi = 1, mu = 0, beta = 1)

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

       n: the number of observations. 

       p: a numeric vector of probabilities. 

       q: a numeric vector of quantiles. 

       x: a numeric vector of quantiles. 

xi, mu, beta: 'xi' is the shape parameter,  'mu' the location
          parameter, and 'beta' is the scale parameter. 

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

     *Generalized Pareto Distribution:* 

      Compute density, distribution function, quantile function and 
     generates random variates for the Generalized Pareto Distribution.

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

     All values are numeric vectors: 
      'd*' returns the density, 
      'p*' returns the probability, 
      'q*' returns the quantiles, and 
      'r*' generates random deviates. 

_A_u_t_h_o_r(_s):

     Alec Stephenson for the functions from R's 'evd' package, 
      Diethelm Wuertz for this R-port.

_R_e_f_e_r_e_n_c_e_s:

     Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); _Modelling
     Extremal Events_, Springer.

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

     'gevFit', 'gpdFit'.

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

     ## *gpd  -
        xmpExtremes("\nStart: Simulate GPD Distributed sample >")
        par(mfrow = c(2, 2))
        r = rgpd(n = 1000, xi = 1/4)
        plot(r, type = "l", main="GPD Series")
        
     ## Plot empirical density and compare with true density:
     ## Omit values greater than 500 from plot
        xmpExtremes("\nNext: Plot Empirical and True Density >")
        hist(r, n = 50, probability = TRUE, xlab = "r", 
          xlim = c(-5, 5), ylim = c(0, 1.1), main = "Density")
        x = seq(-5, 5, by = 0.01)
        lines(x, dgpd(x, xi = 1/4), col = "steelblue3")
        
     ## Plot df and compare with true df:
        xmpExtremes("\nNext: Plot Empirical and True Probability >")
        plot(sort(r), (1:length(r)/length(r)), 
          xlim = c(-3, 6), ylim = c(0, 1.1),
              cex = 0.5, ylab = "p", xlab = "q", main = "Probability")
        q = seq(-5,5, by=0.1)
        lines(q, pgpd(q, xi = 1/4), col = "steelblue3")
        
     ## Compute quantiles, a test:
        xmpExtremes("\nNext: Compute Quantiles >")
        qgev(pgev(seq(-5, 5, 0.25), xi = 1/4 ), xi = 1/4) 

