pickands             package:smoothtail             R Documentation

_C_o_m_p_u_t_e _o_r_i_g_i_n_a_l _a_n_d _s_m_o_o_t_h_e_d _v_e_r_s_i_o_n _o_f _P_i_c_k_a_n_d_s' _e_s_t_i_m_a_t_o_r

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

     Given an ordered sample of either exceedances or upper order
     statistics which is to be modeled using a GPD, this  function
     provides Pickands' estimator of the shape parameter gamma in
     [-1,0].  Precisely, for k=4, ..., n


 hat gamma^k_{rm{Pick}} = frac{1}{log 2} log Bigl(frac{H^{-1}((n-r_k(H)+1)/n)-H^{-1}((n-2r_k(H) +1)/n)}{H^{-1}((n-2r_k(H) +1)/n)-H^{-1}((n-4r_k(H)+1)/n)} Bigr)


     for $H$ either the empirical or the distribution function hat F_n
     based on the log-concave density  estimator and


                      r_k(H) = lfloor k/4 rfloor


     if H is the empirical distribution function and


                            r_k(H) = k / 4


     if H = hat F_n.

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

     pickands(x)

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

       x: Sample of strictly increasing observations.

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

     n x 3 matrix with columns: indices k, Pickands' estimator using
     the smoothing method, and the ordinary Pickands' estimator based
     on the order statistics.

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

     Kaspar Rufibach (maintainer), kaspar.rufibach@freesurf.ch, 
      <URL: http://www.stanford.edu/~kasparr> 

     Samuel Mueller, mueller@maths.uwa.edu.au, 
      <URL: http://www.maths.uwa.edu.au/Members/mueller>

     Kaspar Rufibach acknowledges support by the Swiss National Science
     Foundation SNF, <URL: http://www.snf.ch>

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

     Mueller, S. and Rufibach K. (2006). Smooth tail index estimation.
     Preprint, available at <URL:
     http://arxiv.org/abs/math.ST/0612140>.

     Pickands, J. (1975). Statistical inference using extreme order
     statistics. _Annals of Statistics_ *3*, 119-131.

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

     Other approaches to estimate gamma based on the fact that the
     density is log-concave, thus  gamma in [-1,0], are available as
     the functions 'falk', 'falkMVUE'.

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

     # generate ordered random sample from GPD
     set.seed(1977)
     n <- 20
     gam <- -0.75
     x <- rgpd(n, gam)

     # compute tail index estimators
     pickands(x)

