extRemes (version 2.0-12)

atdf: Auto-Tail Dependence Function

Description

Computes (and by default plots) estimates of the auto-tail dependence function(s) (atdf) based on either chi (rho) or chibar (rhobar), or both.

Usage

atdf(x, u, lag.max = NULL, type = c("all", "rho", "rhobar"), plot = TRUE,
    na.action = na.fail, ...)
# S3 method for atdf
plot(x, type = NULL, ...)

Arguments

x

For atdf: a univariate time series object or a numeric vector. For the plot method function, a list object of class “atdf”.

u

numeric between 0 and 1 (non-inclusive) determining the level F^(-1)(u) over which to compute the atdf. Typically, this should be close to 1, but low enough to incorporate enough data.

lag.max

The maximum lag for which to compute the atdf. Default is 10*log10(n), where n is the length of the data. Will be automatically limited to one less than the total number of observations in the series.

type

character string stating which type of atdf to calculate/plot (rho, rhobar or both). If NULL the plot method function will take the type to be whatever was passed to the call to atdf. If “all”, then a 2 by 1 panel of two plots are graphed.

plot

logical, should the plot be made or not? If TRUE, output is returned invisibly. If FALSE, output is returned normally.

na.action

function to be called to handle missing values.

…

Further arguments to be passed to the plot method function or to plot. Note that if main, xlab or ylab are used with type “all”, then the labels/title will be applied to both plots, which is probably not desirable.

Value

A list object of class “atdf” is returned with components:

call

The function calling string.

type

character naming the type of atdf computed.

series

character string naming the series used.

lag

numeric vector giving the lags used.

atdf

numeric vector or if type is “all”, two-column matrix giving the estimated auto-tail dependence function values.

The plot method functoin does not return anything.

Details

The tail dependence functions are those described in, e.g., Reiss and Thomas (2007) Eq (2.60) for "chi" and Eq (13.25) "chibar", and estimated by Eq (2.62) and Eq (13.28), resp. See also, Sibuya (1960) and Coles (2001) sec. 8.4, as well as other texts on EVT such as Beirlant et al. (2004) sec. 9.4.1 and 10.3.4 and de Haan and Ferreira (2006).

Specifically, for two series X and Y with associated df's F and G, chi, a function of u, is defined as

chi(u) = Pr[Y > G^(-1)(u) | X > F^(-1)(u)] = Pr[V > u | U > u],

where (U,V) = (F(X),G(Y))--i.e., the copula. Define chi = limit as u goes to 1 of chi(u).

The coefficient of tail dependence, chibar(u) was introduced by Coles et al. (1999), and is given by

chibar(u) = 2*log(Pr[U > u])/log(Pr[U > u, V > u]) - 1.

Define chibar = limit as u goes to 1 of chibar(u).

The auto-tail dependence function using chi(u) and/or chibar(u) employs X against itself at different lags.

The associated estimators for the auto-tail dependence functions employed by these functions are based on the above two coefficients of tail dependence, and are given by Reiss and Thomas (2007) Eq (2.65) and (13.28) for a lag h as

rho.hat(u, h) = sum(min(x_i, x_i+h) > sort(x)[floor(n*u)])/(n*(1-u)) [based on chi]

and

rhobar.hat(u, h) = 2*log(1 - u)/log(sum(min(x_i,x_i+h) > sort(x)[floor(n*u)])/(n - h)) - 1.

Some properties of the above dependence coefficients, chi(u), chi, and chibar(u) and chibar, are that 0 <= chi(u), chi <= 1, where if X and Y are stochastically independent, then chi(u) = 1 - u, and chibar = 0. If X = Y (perfectly dependent), then chi(u) = chi = 1. For chibar(u) and chibar, we have that -1 <= chibar(u), chibar <= 1. If U = V, then chibar = 1. If chi = 0, then chibar < 1 (tail independence with chibar determining the degree of dependence).

References

Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004) Statistics of Extremes: Theory and Applications. Chichester, West Sussex, England, UK: Wiley, ISBN 9780471976479, 522pp.

Coles, S. (2001) An introduction to statistical modeling of extreme values, London, U.K.: Springer-Verlag, 208 pp.

Coles, S., Heffernan, J. E., and Tawn, J. A. (1999) Dependence measures for extreme value analyses. Extremes, 2, 339--365.

de Haan, L. and Ferreira, A. (2006) Extreme Value Theory: An Introduction. New York, NY, USA: Springer, 288pp.

Reiss, R.-D. and Thomas, M. (2007) Statistical Analysis of Extreme Values: with applications to insurance, finance, hydrology and other fields. Birkh\"auser, 530pp., 3rd edition.

Sibuya, M. (1960) Bivariate extreme statistics. Ann. Inst. Math. Statist., 11, 195--210.

See Also

acf, pacf, taildep, taildep.test

Examples

# NOT RUN {
z <- arima.sim(n = 63, list(ar = c(0.8897, -0.4858), ma = c(-0.2279, 0.2488)),
               sd = sqrt(0.1796))
hold <- atdf(z, 0.8, plot=FALSE)
par(mfrow=c(2,2))
acf(z, xlab="")
pacf(z, xlab="")
plot(hold, type="chi")
plot(hold, type="chibar")

y <- cbind(z[2:63], z[1:62])
y <- apply(y, 1, max)
hold2 <- atdf(y, 0.8, plot=FALSE)
par(mfrow=c(2,2))
acf(y, xlab="")
pacf(y, xlab="")
plot(hold2, type="chi")
plot(hold2, type="chibar")

# }
# NOT RUN {
data(Fort)
atdf(Fort[,5], 0.9)

data(Tphap)
atdf(Tphap$MaxT, 0.8)

data(PORTw)
atdf(PORTw$TMX1, u=0.9)
atdf(PORTw$TMX1, u=0.8)
# }