awstindex {aws} R Documentation

## 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` `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.

### 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.

### Value

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

### Author(s)

Joerg Polzehl, polzehl@wias-berlin.de

### 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 `aws`, `laws`

### Examples

```###
###   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")
```

[Package aws version 1.3-0 Index]