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")

[Package *aws* version 1.3-0 Index]