awstindex: Tail index estimation

Description

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.

Usage

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

Arguments

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

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

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"

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.

`Value`

The returned object is a list with components
tindexEstimated tail-index
intensityEstimates of the intensity in the exponential model
yValues of y
callactual function call

`Details`

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.

`References`

{    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
 }

`See Also`

SEE ALSO aws, laws

`Examples`

Run this code###
###   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")
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