ETDF {AoE} | R Documentation |
Computes the empirical tail dependence function based on a bivariate sample (X_1, Y_1), ..., (X_n, Y_n).
ETDF(data.x, data.y, data = NULL, v = c(1, 1), k, method = "empirical", plot = TRUE)
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. |
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.
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.
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.
# 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)