awstindex {aws} | R Documentation |
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@wias-berlin.de
### ### 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")