abvnonpar                package:evd                R Documentation

_N_o_n-_p_a_r_a_m_e_t_r_i_c _E_s_t_i_m_a_t_e_s _f_o_r _D_e_p_e_n_d_e_n_c_e _F_u_n_c_t_i_o_n_s _o_f _t_h_e
_B_i_v_a_r_i_a_t_e _E_x_t_r_e_m_e _V_a_l_u_e _D_i_s_t_r_i_b_u_t_i_o_n

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

     Calculate or plot non-parametric estimates for the dependence
     function A of the bivariate extreme value distribution.

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

     abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL,
         method = c("cfg", "pickands", "deheuvels", "hall", "tdo"),
         convex = FALSE, wf = function(t) t, plot = FALSE,
         add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, xlim = c(0, 1),
         ylim = c(0.5, 1), xlab = "", ylab = "", ...)

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

       x: A vector of values at which the dependence function is
          evaluated (ignored if plot or add is 'TRUE'). A(1/2) is
          returned by default since it is often a useful summary of
          dependence.

    data: A matrix or data frame with two columns, which may contain
          missing values.

nsloc1, nsloc2: A data frame with the same number of rows as 'data',
          for linear modelling of the location parameter on the
          first/second margin. The data frames are treated as covariate
          matrices, excluding the intercept. A numeric vector can be
          given as an alternative to a single column data frame.

  method: The estimation method (see *Details*). The '"cfg"' method is
          used by default.

  convex: Logical; take the convex minorant?

      wf: The weight function used in the '"cfg"' method (see
          *Details*). The function must be vectorized.

    plot: Logical; if 'TRUE' the function is plotted. The x and y
          values used to create the plot are returned invisibly. If
          'plot' and 'add' are 'FALSE' (the default), the arguments
          following 'add' are ignored.

     add: Logical; add to an existing plot? The existing plot should
          have been created using either 'abvnonpar' or 'abvpar', the
          latter of which plots (or calculates) the dependence function
          for a number of parametric models.

lty, blty: Function and border line types. Set 'blty' to zero to omit
          the border.

     lwd: Line width.

     col: Line colour.

xlim, ylim: x and y-axis limits.

xlab, ylab: x and y-axis labels.

     ...: Other high-level graphics parameters to be passed to 'plot'.

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

     The dependence function  A() of the bivariate extreme value
     distribution is defined in 'abvpar'. Non-parametric estimates are
     constructed as follows. Suppose (z_{i1},z_{i2}) for i=1,...,n are
     n bivariate observations that are passed using the 'data'
     argument. The marginal parameters are estimated (under the
     assumption of independence) and the data is transformed using

           y_{i1} = {1 + s'_1(z_{i1}-a'_1)/b'_1}^(-1/s'_1)

     and

           y_{i2} = {1 + s'_2(z_{i2}-a'_2)/b'_2}^(-1/s'_2)

     for i = 1,...,n, where (a'_1,b'_1,s'_1) and (a'_2,b'_2,s'_2) are
     the maximum likelihood estimates for the location, scale and shape
     parameters on the first and second margins. If 'nsloc1' or
     'nsloc2' are given, the location parameters may depend on i (see
     'fgev').

     Five different estimators of the dependence function can be
     implemented. They are defined (on 0 <= w <= 1) as follows.

     'method = "cfg"' (Caperaa, Fougeres and Genest, 1997)

 A_c(w) = exp{ [1-p(w)] integral_0^w (H(x) - x)/[x(1-x)] dx - p(w) integral_w^1 (H(x) - x)/[x(1-x)] dx }


     'method = "pickands"' (Pickands, 1981)

        A_p(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}/(1-w)]}


     'method = "deheuvels"' (Deheuvels, 1991)

 A_d(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}(1-w)] - w sum_{i=1}^n y_{i1} - (1-w) sum_{i=1}^n y_{i2} + n}


     'method = "hall"' (Hall and Tajvidi, 2000)

 A_h(w) = n (sum_{i=1}^n min[y_{i1}/(by_1 w), y_{i2}/(by_2 (1-w))])^{-1}


     'method = "tdo"' (Tiago de Oliveira, 1997)

 A_t(w) = 1 - 1/(1 + log n) sum_{i=1}^n min[w/(1 + n y_{i1}), (1 - w)/(1 + n y_{i2})]


     In the estimator A_h(), by_j = (sum_{i=1}^n y_{ij})/n for j = 1,2.
     In the estimator A_c(), H(x) is the empirical distribution
     function of x_1,...,x_n, where x_i = y_{i1} / (y_{i1} + y_{i2})
     for i = 1,...,n, and p(w) is any bounded function on [0,1], which
     can be specified using the argument 'wf'. By default 'wf' is the
     identity function.

     Let A_n() be any estimator of A(). The constraint A_n(0) = A_n(1)
     = 1 is satisfied by A_d(), A_t() and A_h(), and by A_c() when p(0)
     = 0 and p(1) = 1. None of the estimators satisfy max(w,1-w) <=
     A_n(w) <= 1 for all 0 <= w <= 1. An obvious modification is

                A'_n(w) = min(1, max{A_n(w), w, 1-w}).

     This modification is always implemented.

     A_t(w) is the only estimator that is convex. Convex estimators can
     be derived from other methods by taking the convex minorant, which
     can be achieved by setting 'convex' to 'TRUE'.

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

     'abvnonpar' calculates or plots a non-parametric estimate of the
     dependence function of the bivariate extreme value distribution.

_N_o_t_e:

     Appendix A of the User's Guide contains a short simulation study
     that compares the estimators defined above. The estimators A_p(),
     A_d() and A_h() are very similar, and may not be distinguishable
     when plotted.

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

     Caperaa, P. Fougeres, A.-L. and Genest, C. (1997) A non-parametric
     estimation procedure for bivariate extreme value copulas.
     _Biometrika_, *84*, 567-577.

     Deheuvels, P. (1991) On the limiting behaviour of the Pickands
     estimator  for bivariate extreme-value distributions. _Statist.
     Probab. Letters_, *12*, 429-439.

     Hall, P. and Tajvidi, N. (2000) Distribution and
     dependence-function estimation for bivariate extreme-value
     distributions. _Bernoulli_, *6*, 835-844.

     Pickands, J. (1981) Multivariate extreme value distributions.
     _Proc. 43rd Sess. Int. Statist. Inst._, *49*, 859-878.

     Tiago de Oliveira, J. (1997) _Statistical Analysis of Extremes._
     Pendor.

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

     'abvpar', 'atvnonpar', 'fgev'

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

     bvdata <- rbvevd(100, dep = 0.7, model = "log")
     abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE)
     abvnonpar(data = bvdata, method = "d", plot = TRUE)

     M1 <- fitted(fbvevd(bvdata, model = "log"))
     abvpar(dep = M1["dep"], model = "log", plot = TRUE)
     abvnonpar(data = bvdata, add = TRUE, lty = 2)

