ETDF {AoE}R Documentation

Empirical Tail Dependence Function

Description

Computes the empirical tail dependence function based on a bivariate sample (X_1, Y_1), ..., (X_n, Y_n).

Usage

ETDF(data.x, data.y, data = NULL, v = c(1, 1), k, 
                        method = "empirical", plot = TRUE)

Arguments

data.x, data.y Numeric vectors containing the data X_1, ..., X_n and Y_1, ..., Y_n, respectively.
data Alternatively, the data may be provided in the form of a n-by-2 matrix. If provided, then the arguments data.x and data.y are ignored.
v The point in which the empirical tail dependence function is to be computed.
k An numeric vector of values for k in the definition of the empirical tail dependence function; see ‘Details’.
method The estimation method, specified by a string. Currently, this argument is ignored since only the empirical method "empirical" is implemented.
plot If TRUE (the default), the result will be plotted.

Details

The empirical tail dependence function for a bivariate sample (X_1, Y_1), ..., (X_n, Y_n) is defined by

hat{l}(x, y) = (1/k) * sum_{i=1}^n I(R_i >= n+1-k*x or S_i >= n+1-k*y)

where x, y >= 0, where R_i and S_i are the ranks of the data, and where 0 < k < n is a tuning parameter. The elements of the input vector v correspond to the values of x and y.

The function is an estimate of the (stable) tail dependence function

l(x, y) = lim_{s -> 0} (1 - C(1 - s*x, 1 - s*y))/s

where C is the copula of the underlying distribution. In order for the estimator to be consistent, we need k = k(n) with k to infinity and k/n to zero.

A useful special case is when (x, y) = (1, 1), for lambda = 2 - l(1, 1) is the coefficient of tail dependence:

lambda = lim_{s -> 0} Pr(F_X(X) > 1-s | F_Y(Y) > 1-s)

In particular, l(1, 1) = 1 corresponds to asymptotic complete dependence, while l(1, 1) = 2 corresponds to asymptotic independence. More generally, low (high) values of l(1,1) indicate strong (weak) tail dependence.

Another special case is when y = 1-x, yielding the Pickands dependence function

A(x) = l(x, 1-x)

for 0 <= x <= 1.

Value

A numeric vector of length length(k), the elements being the corresponding estimates of the tail dependence function at the point specified by v. The result is returned invisibly.

References

Drees, H. and Huang, X. (1998). Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function. Journal of Multivariate Analysis 64, 25-47.

See Also

AngularMeasure, PickandsDF

Examples

# The bivariate normal distribution 
# with arbitrary correlation not equal to one 
# has an asymptotically independent upper tail:
ETDF(data = rbivnorm(1e5, cor = 0.9), k = 10:100)

# The Loss-ALAE data seem to exhibit asymptotic dependence:
data(Loss, ALAE)
ETDF(data.x = Loss, data.y = ALAE, k = 10:100)

[Package AoE version 1.0.1 Index]