POT (version 1.1-7)

tsdep.plot: Diagnostic for Dependence within Time Series Extremes

Description

A diagnostic tool to assess for short range asymptotic dependence within a stationary time series.

Usage

tsdep.plot(data, u, …, xlab, ylab, n.boot = 100, show.lines = TRUE,
lag.max, ci = 0.95, block.size = 5 * lag.max, angle = 90, arrow.length =
0.1)

Arguments

data

The time series observations.

u

The threshold.

…

Optional arguments to be passed to the plot function.

xlab,ylab

The x and y-axis labels.

n.boot

Numeric. The number of replicates to compute the bootstrap confidence interval.

show.lines

Logical. If TRUE (the default), the theoretical lines for the asymptotic dependence and ``near'' independence are drawn.

lag.max

The maximum lag to be explored - may be missing.

ci

The level for the bootstrap confidence interval. The default is the 95% confidence interval.

block.size

The size for the contiguous bootstrap approach.

angle

The angle at the end of the error bar. If 0, error bars are only segments.

arrow.length

The length to be passed in the function arrows.

Value

This function plot the \(\Lambda_\tau\) statictics against the lag. Bootstrap confidence intervals are also drawn. The function returns invisibly this statistic and the confidence bounds.

Details

Let X_t be a stationary sequence of unit Frechet random variables. By stationarity, the joint survivor function \(\overline{F}_\tau(\cdot, \cdot)\) of \((X_t, X_{t+\tau})\) does not depend on \(t\).

One parametric representation for \(\overline{F}_\tau(\cdot, \cdot)\) is given by $$\overline{F}_\tau(s,s)=L_\tau(s) s^{-1/\eta_\tau}$$ for some parameter \(\eta_\tau \in (0,1]\) and a slowly varying function \(L_\tau\).

The \(\Lambda_\tau\) statistic is defined by $$\Lambda_\tau = 2 \eta_\tau - 1$$ This statistic belongs to (-1,1] and is a measure of extremal dependence. \(\Lambda_\tau = 1\) corresponds to asymptotic dependence, \(0 < \Lambda_\tau < 1\) to positive extremal association, \(\Lambda_\tau = 0\) to ``near'' independence and \(\Lambda_\tau < 0\) to negative extremal association.

References

Ledford, A. and Tawn, J. (2003) Diagnostics for dependence within time series extremes. L. R. Statist. Soc. B. 65, Part 2, 521--543.

Ledford, A. and Tawn, J (1996) Statistics for near independence in multivariate extreme values. Biometrika 83 169--187.

See Also

chimeas, tailind.test

Examples

# NOT RUN {
##An independent case
tsdep.plot(runif(5000), u = 0.95, lag.max = 5)
##Asymptotic dependence
mc <- simmc(5000, alpha = 0.2)
tsdep.plot(mc, u = 0.95, lag.max = 5)
# }