awstindex                package:aws                R Documentation

_T_a_i_l _i_n_d_e_x _e_s_t_i_m_a_t_i_o_n

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

     The function finds a pareto-approximation of the tail of a
     univariate  distribution and estimates the parameter in this
     pareto-approximation. The construction is similar to the
     Hill-estimator. The number of largest observations used in the
     estimate is chosen adaptively.

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

     awstindex(y, qlambda = NULL, eta = 0.5, lkern = "Triangle", hinit = 1, 
               hincr = 1.25, hmax = 1000, graph = FALSE, symmetric = FALSE)

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

       y: 'y' contains the observed values at location 'x'.  In case of
          'x=NULL' (second parameter) 'y' is assumed to be  observed on
          a one, two or three-dimensional grid. The dimension of  'y'
          determines if one, two or three-dimensional AWS is used.

 qlambda: 'qlambda' determines the scale parameter 'qlambda'  for the
          stochastic penalty. The scaling parameter in the stochastic 
          penalty 'lambda' is choosen as the 'qlambda'-quantile of a
          Chi-square-distribution with number of parameters in the
          polynomial  model as degrees of freedom. If 'qlambda=NULL' a
          standard value  depending on 'model' and 'symmetric' is
          choosen. 

     eta: 'eta' is a memory parameter used to stabilize the procedure. 
          'eta'  has to be between '0' and '1', with  'eta=.5' being
          the default. 

   lkern: 'lkern' determines the location kernel to be used. Options 
          are '"Uniform"', '"Triangle"', '"Quadratic"',  '"Cubic"' and
          '"Exponential"'. Default is '"Triangle"'.  The Kernel
          operates on the squared distance, so '"Triangle"' corresponds
          to the use of an Epanechnikov kernel in kernel smoothing. 
          '"Exponential"' requires larger values of 'hmax' and 
          therefore more iterations to reach comparable results. 

   hinit: 'hinit' Initial bandwidth for the location penalty. 
          Appropriate value is choosen in case of 'hinit==NULL'  

   hincr: 'hincr' 'hincr^(1/d)', with 'd' the  dimensionality of the
          design, is used as a factor to increase the  bandwidth
          between iterations. Defauts to 'hincr=1.2'

    hmax: 'hmax' Maximal bandwidth to be used. Determines the  number
          of iterations and is used as the stopping rule. 

   graph: 'graph' if 'TRUE' results are displayed after each  iteration
          step. 

symmetric: If 'symmetric==TRUE' the stochastic penalty is symmetrized,
          i.e. '(sij + sji)/lambda' is used instead of  'sij/lambda'.
          See references for details.

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

     From the data 'y' an descending order statistics 'yn <-
     order(y)[n:1]' is computed  and transformed observations 'x <-
     (1:(n-1))*yn[-n]/yn[-1]' are defined. The transformed observations
     are assumed to follow an inhomogenious exponential model. Adaptive
     Weights Smoothing, i.e. function 'laws' with parameter
     'model="Exponential"', is used  to construct an inhomogenious
     intensity estimate. The estimated tail index is the estimated
     intensity in the left-most point, corresponding to the largest
     observation in the sample. This estimate is similar to the
     Hill-estimate computed from the 'k' largest observations with 'k'
     approximately equal to the sum of weights used for estimating the
     tail index by AWS. See Section 8 in Polzehl and Spokoiny (2002)
     for details.

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

     The returned object is a list with components 

 tindex : Estimated tail-index

intensity : Estimates of the intensity in the exponential model

      y : Values of 'y'

   call : actual function call

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

     Joerg Polzehl, polzehl@wias-berlin.de 

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

  Polzehl, J. and Spokoiny, V. (2002).  _Local likelihood modelling by
     adaptive weights smoothing_,  WIAS-Preprint 787 

  Polzehl, J. and Spokoiny, V. (2000). _Adaptive Weights Smoothing with
     applications to image restoration_, J.R.Statist.Soc. B, 62, Part
     2, pp.335-354 

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

     SEE ALSO 'aws', 'laws'

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

     ###
     ###   Estimate the tail-index of a cauchy distribution
     ###   absolute values can be used because of the symmetry of centered cauchy
     ###
     set.seed(1)
     n <- 500
     x <- rcauchy(n)
     tmp <- awstindex(abs(x),hmax=n)
     tmp$tindex
     ###
     ###   now show the segmentation generated by AWS 
     ###
     plot(tmp$intensity[1:250],type="l")

