GevMdaEstimation          package:fExtremes          R Documentation

_G_e_n_e_r_a_l_i_z_e_d _E_x_t_r_e_m_e _V_a_l_u_e _M_o_d_e_l_l_i_n_g

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

     A collection and description functions to estimate  the parameters
     of the GEV distribution. To model the GEV three types of
     approaches for parameter  estimation are provided: Maximum
     likelihood estimation, probability weighted moment method, and
     estimation by the MDA approach. MDA includes functions for the
     Pickands, Einmal-Decker-deHaan,  and Hill estimators together with
     several plot  variants. 

     Maximum Domain of Attraction estimators:

       'hillPlot'     shape parameter and Hill estimate of the tail index,
       'shaparmPlot'  variation of shape parameter with tail depth.

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

     hillPlot(x, start = 15, ci = 0.95, 
         doplot = TRUE, plottype = c("alpha", "xi"), labels = TRUE, ...)
     shaparmPlot(x, p = 0.01*(1:10), xiRange = NULL, alphaRange = NULL,
         doplot = TRUE, plottype = c("both", "upper"))
         
     shaparmPickands(x, p = 0.05, xiRange = NULL,  
         doplot = TRUE, plottype = c("both", "upper"), labels = TRUE, ...) 
     shaparmHill(x, p = 0.05, xiRange = NULL,  
         doplot = TRUE, plottype = c("both", "upper"), labels = TRUE, ...)
     shaparmDEHaan(x, p = 0.05, xiRange = NULL,  
         doplot = TRUE, plottype = c("both", "upper"), labels = TRUE, ...)

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

alphaRange, xiRange: [saparmPlot] - 
           plotting ranges for 'alpha' and 'xi'. By default the values
          are automatically selected. 

      ci: [hillPlot] - 
           probability for asymptotic confidence band; for no
          confidence band set 'ci' to zero. 

  doplot: a logical. Should the results be plotted? 
           [shaparmPlot] - 
           a vector of logicals of the same lengths as tails defining
          for wich tail depths plots should be created, by default
          plots will be generated for a tail depth of 5 percent. By
          default 'c(FALSE, FALSE, FALSE, FALSE,  TRUE, FALSE, FALSE,
          FALSE, FALSE, FALSE)'. 

  labels: [hillPlot] - 
           whether or not axes should be labelled. 

plottype: [hillPlot] - 
           whether 'alpha', 'xi' (1/alpha) or 'quantile' (a quantile
          estimate) should be plotted. 

       p: [qgev] - 
           a numeric vector of probabilities. [hillPlot] - 
           probability required when option 'quantile' is chosen. 

   start: [hillPlot] - 
           lowest number of order statistics at which to plot a point. 

       x: [dgev][devd] - 
           a numeric vector of quantiles. 
           [gevFit] - 
           data vector. In the case of 'method="mle"' the
          interpretation  depends on the value of block: if no block
          size is specified then  data are interpreted as block maxima;
          if block size is set, then data are interpreted as raw data
          and block maxima are calculated.  
           [hillPlot][shaparmPlot] - 
           the data from which to calculate the shape parameter, a
          numeric vector. 
           [print][plot] - 
           a fitted object of class '"gevFit"'. 

     ...: [gevFit] - 
           control parameters optionally passed to the  optimization
          function. Parameters for the optimization function are passed
          to components of the 'control' argument of 'optim'.  
           [hillPlot] - 
           other graphics parameters. 
           [plot][summary] - 
           arguments passed to the plot function. 

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

     *Parameter Estimation:* 

      'gevFit' and 'gumbelFit' estimate the parameters either  by the
     probability weighted moment method, 'method="pwm"' or  by maximum
     log likelihood estimation 'method="mle"'. The  summary method
     produces diagnostic plots for fitted GEV or Gumbel  models. 

     *Methods:* 

      'print.gev', 'plot.gev' and 'summary.gev' are print, plot, and
     summary methods for a fitted object of class  'gev'. Concerning
     the summary method, the data are  converted to unit exponentially
     distributed residuals under null  hypothesis that GEV fits. Two
     diagnostics for iid exponential data  are offered. The plot method
     provides two different residual plots  for assessing the fitted
     GEV model. Two diagnostics for  iid exponential data are offered.  

     *Return Level Plot:* 

      'gevrlevelPlot' calculates and plots the k-block return level 
     and 95% confidence interval based on a GEV model for block maxima,
      where 'k' is specified by the user. The k-block return level  is
     that level exceeded once every 'k' blocks, on average. The  GEV
     likelihood is reparameterized in terms of the unknown return 
     level and profile likelihood arguments are used to construct a 
     confidence interval.  

     *Hill Plot:* 

      The function 'hillPlot' investigates the shape parameter and 
     plots the Hill estimate of the tail index of heavy-tailed data, or
      of an associated quantile estimate. This plot is usually
     calculated  from the alpha perspective. For a generalized Pareto
     analysis of  heavy-tailed data using the 'gpdFit' function, it
     helps to  plot the Hill estimates for 'xi'.  

     *Shape Parameter Plot:* 

      The function 'shaparmPlot' investigates the shape parameter and 
     plots for the upper and lower tails the shape parameter as a
     function  of the taildepth. Three approaches are considered, the
     _Pickands_  estimator, the _Hill_ estimator, and the
     _Decker-Einmal-deHaan_ estimator.

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

     'gevSim' 
      returns a vector of data points from the simulated series. 

     'gevFit' 
      returns an object of class 'gev' describing the fit. 

     'print.summary' 
      prints a report of the parameter fit. 

     'summary' 
      performs diagnostic analysis. The method provides two different 
     residual plots for assessing the fitted GEV model.   

     'gevrlevelPlot' 
      returns a vector containing the lower 95% bound of the confidence
      interval, the estimated return level and the upper 95% bound.  

     'hillPlot' 
      displays a plot. 

     'shaparmPlot'  
      returns a list with one or two entries, depending on the
     selection of the input variable 'both.tails'. The two  entries
     'upper' and 'lower' determine the position of  the tail. Each of
     the two variables is again a list with entries  'pickands',
     'hill', and 'dehaan'. If one of the  three methods will be
     discarded the printout will display zeroes.

_N_o_t_e:

     *GEV Parameter Estimation:* 

      If method '"mle"' is selected the parameter fitting in 'gevFit' 
     is passed to the internal function 'gev.mle' or 'gumbel.mle'
     depending on the value of 'gumbel', 'FALSE' or 'TRUE'. On the
     other hand, if method '"pwm"' is selected the parameter  fitting
     in 'gevFit' is passed to the internal function  'gev.pwm' or
     'gumbel.pwm' again depending on the value of  'gumbel', 'FALSE' or
     'TRUE'.

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

     Alec Stephenson for R's 'evd' and 'evir' package, and 
      Diethelm Wuertz for this R-port.

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

     Coles S. (2001); _Introduction to Statistical Modelling of Extreme
     Values_, Springer.

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

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

       
     ## Load Data:
        x = as.timeSeries(data(danishClaims))
        colnames(x) <- "Danish"
        head(x)
        
     ## hillPlot -
        # Hill plot of heavy-tailed Danish fire insurance data 
        par(mfrow = c(1, 1))
        hillPlot(x, plottype = "xi")
        grid()

