smoothtail-package        package:smoothtail        R Documentation

_S_m_o_o_t_h _E_s_t_i_m_a_t_i_o_n _o_f _G_P_D _S_h_a_p_e _P_a_r_a_m_e_t_e_r

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

     Given independent and identically distributed observations X_1 <
     ... < X_n from a  Generalized Pareto distribution with shape
     parameter gamma in [-1,0], this package offers three  methods to
     compute estimates of gamma. The estimates are based on the
     principle of replacing the order  statistics X_{(1)}, ..., X_{(n)}
     of the sample by quantiles hat X_{(1)}, ..., hat X_{(n)} of the
     distribution function hat F_n based on the  log-concave density
     estimator hat f_n. This procedure is justified by the fact that
     the GPD density is  log-concave for gamma in [-1,0].

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


       Package:  smoothtail
       Type:     Package
       Version:  1.1.0
       Date:     2007-08-17
       License:  GPL version 2 or newer

     Use this package to estimate the shape parameter gamma of a
     Generalized Pareto Distribution (GPD). In  extreme value theory,
     gamma is denoted tail index. We offer three new estimators, all
     based on the fact  that the density function of the GPD is
     log-concave if gamma in [-1,0]. The functions for estimation of
     the tail index are:

     'pickands' 
      'falk'
      'falkMVUE'

     Additionally, functions for density, distribution function,
     quantile function and random number generation for a GPD with
     location parameter 0, shape parameter gamma and scale parameter
     sigma are provided:

     'dgpd' 
      'pgpd'
      'qgpd'
      'rgpd'.

     This package depends on the package 'logcondens' for estimation of
     a log-concave density.

     Let us shortly clarify what we mean with log-concave density
     estimation. Suppose we are given an ordered sample Y_1 < ... < Y_n
     of i.i.d. random variables having density function f, where f =
     exp varphi for a concave function varphi : [-infty, infty) to R.
     Following the development in  Duembgen and Rufibach (2007), it is
     then possible to get an estimator hat f_n = exp hat varphi_n  of f
     via the maximizer hat varphi_n of


    L(varphi) =  sum_{i=1}^n varphi(Y_i) -  int exp varphi (t) d t


     over all concave functions varphi. It turns out that hat varphi_n
     is piecewise linear, with  knots only at (some of the) observation
     points. Therefore, the infinite-dimensional optimization problem
     of finding  the function hat varphi_n boils down to a finite
     dimensional problem of finding the vector (hat
     varphi_n(Y_1),...,hat varphi(Y_n)).  How to solve this problem is
     described in Rufibach (2006, 2007) and in a more general setting
     in Duembgen, Huesler, and Rufibach (2007). The distribution
     function based on hat f_n is defined as


               hat F_n(x) = int_{Y_1}^x hat f_n(t) d t


     for x a real number. The definition of hat F_n is justified by the
     fact that hat F_n(Y_1) = 0.

_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:

     Duembgen, L. and Rufibach, K. (2007). Maximum likelihood
     estimation of a log-concave density and its distribution function:
     basic properties and uniform consistency.  Preprint, IMSV,
     University of Bern.

     Duembgen, L., Huesler, A. and Rufibach, K. (2007) Active set and
     EM algorithms for log-concave densities based on complete and
     censored data. Technical report 61, IMSV, Univ. of Bern, available
     at <URL: http://arxiv.org/abs/0707.4643>.

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

     Mueller, S. and Rufibach K. (2006b). On the max-domain of
     attraction of distributions with log-concave densities. Submitted.

     Rufibach K. (2006) _Log-concave Density Estimation and Bump
     Hunting for i.i.d. Observations._ PhD Thesis, University of Bern,
     Switzerland and Georg-August University of Goettingen, Germany,
     2006. 
      Available at <URL:
     http://www.stub.unibe.ch/download/eldiss/06rufibach_k.pdf>.

     Rufibach, K. (2007) Computing maximum likelihood estimators of a
     log-concave density function. _J. Stat. Comput. Simul._, *77*,
     561-574.

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

     Package 'logcondens'.

_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 known endpoint
     omega <- -1 / gam

     # estimate log-concave density, for illustration purposes
     est <- activeSetLogCon(x)

     # plot distribution functions
     s <- seq(0.01, max(x), by = 0.01)
     plot(0, 0, type = 'n', ylim = c(0, 1), xlim = range(c(x, s))); rug(x)
     lines(s, pgpd(s, gam), type = 'l', col = 2)
     lines(x, 1:n / n, type = 's', col = 3)
     lines(x, est$Fhat, type = 'l', col = 4)
     legend(1, 0.4, c('true', 'empirical', 'estimated'), col = c(2 : 4), lty = 1)

     # compute tail index estimators
     falk.logcon <- falk(x)
     falkMVUE.logcon <- falkMVUE(x, omega)
     pick.logcon <- pickands(x)

     # plot smoothed and unsmoothed estimators versus number of order statistics
     plot(0, 0, type = 'n', xlim = c(0,n), ylim = c(-1, 0.2))
     lines(1:n, pick.logcon[,2], col = 1); lines(1:n, pick.logcon[,3], col = 1, lty = 2)
     lines(1:n, falk.logcon[,2], col = 2); lines(1:n, falk.logcon[,3], col = 2, lty = 2)
     lines(1:n, falkMVUE.logcon[,2], col = 3); lines(1:n, falkMVUE.logcon[,3], col = 3, lty = 2)
     abline(h = gam, lty = 3)
     legend(11, 0.2, c("Pickands", "Falk", "Falk MVUE"), lty = 1, col = 1:6)

