falk               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 _F_a_l_k'_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 Falk's estimator of the shape parameter gamma in [-1,0].
     Precisely,


 hat gamma_{rm{Falk}} = hat gamma_{rm{Falk}}(k, n) = frac{1}{k-1} sum_{j=2}^k log Bigl(frac{X_{(n)}-H^{-1}((n-j+1)/n)}{X_{(n)}-H^{-1}((n-k)/n)} Bigr), ; ; k=3, ... ,n-1


     for $H$ either the empirical or the distribution function based on
     the log-concave density estimator. Note that for any k, hat
     gamma_{rm{Falk}} : R^n to (-infty, 0). If  hat gamma_{rm{Falk}}
     not in [-1,0), then it is likely that the log-concavity assumption
     is violated.

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

     falk(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, Falk's estimator using the
     smoothing method, and the ordinary Falk's 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>.

     Falk, M. (1995). Some best parameter estimates for distributions
     with finite endpoint. _Statistics_, *27*, 115-125.

_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 'pickands', '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
     falk(x)

