# Extremal Index Estimation

### Description

A collection and description of functions to simulate
time series with a known extremal index, and to estimate
the extremal index by four different kind of methods,
the blocks method, the reciprocal mean cluster size
method, the runs method, and the method of Ferro and
Segers.

The functiona are:

`thetaSim` | Simulates a time Series with known theta, |

`blockTheta` | Computes theta from Block Method, |

`clusterTheta` | Computes theta from Reciprocal Cluster Method, |

`runTheta` | Computes theta from Run Method, |

`ferrosegersTheta` | Computes Theta according to Ferro and Seegers, |

`exindexPlot` | Calculate and Plot Theta(1,2,3), |

`exindexesPlot` | Calculate Theta(1,2) and Plot Theta(1). |

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
## S4 method for signature 'fTHETA'
show(object)
thetaSim(model = c("max", "pair"), n = 1000, theta = 0.5)
blockTheta(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
title = NULL, description = NULL)
clusterTheta(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
title = NULL, description = NULL)
runTheta(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
title = NULL, description = NULL)
ferrosegersTheta(x, quantiles = seq(0.950, 0.995, length = 10),
title = NULL, description = NULL)
exindexPlot(x, block = c("monthly", "quarterly"), start = 5, end = NA,
doplot = TRUE, plottype = c("thresh", "K"), labels = TRUE, ...)
exindexesPlot(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
doplot = TRUE, labels = TRUE, ...)
``` |

### Arguments

`block` |
[*Theta] - |

`description` |
a character string which allows for a brief description. |

`doplot` |
a logical, should the results be plotted? |

`labels` |
whether or not axes should be labelled. If set to |

`model` |
[thetaSim] - |

`n` |
[thetaSim] - |

`object` |
an object of class |

`plottype` |
[exindexPlot] - |

`quantiles` |
[exindexesPlot] - |

`start, end` |
[exindexPlot] - |

`theta` |
[thetaSim] - |

`title` |
a character string which allows for a project title. |

`x` |
a 'timeSeries' object or any other object which can be transformed
by the function |

`...` |
additional arguments passed to the plot function. |

### Value

`exindexPlot`

returns a data frame of results with the
following columns: `N`

, `K`

, `un`

, `theta2`

,
and `theta`

. A plot with `K`

on the lower x-axis and
threshold Values on the upper x-axis versus the extremal index
is displayed.

`exindexesPlot`

returns a data.frame with four columns:
`thresholds`

, `theta1`

, `theta2`

, and `theta3`

.
A plot with quantiles on the x-axis and versus the extremal indexes
is displayed.

### Author(s)

Alexander McNeil, for parts of the `exindexPlot`

function, and

Diethelm Wuertz for the `exindexesPlot`

function.

### References

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997);
*Modelling Extremal Events*,
Springer. Chapter 8, 413–429.

### See Also

`hillPlot`

,
`gevFit`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 | ```
## Extremal Index for the right and left tails
## of the BMW log returns:
data(bmwRet)
par(mfrow = c(2, 2), cex = 0.7)
exindexPlot( as.timeSeries(bmwRet), block = "quarterly")
exindexPlot(-as.timeSeries(bmwRet), block = "quarterly")
## Extremal Index for the right and left tails
## of the BMW log returns:
exindexesPlot( as.timeSeries(bmwRet), block = 65)
exindexesPlot(-as.timeSeries(bmwRet), block = 65)
``` |