awstindex {aws}R Documentation

Tail index estimation


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.


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


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.


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.


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


Joerg Polzehl,


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


###   Estimate the tail-index of a cauchy distribution
###   absolute values can be used because of the symmetry of centered cauchy
n <- 500
x <- rcauchy(n)
tmp <- awstindex(abs(x),hmax=n)
###   now show the segmentation generated by AWS 

[Package aws version 1.3-0 Index]