Non-parametric estimators of the STDF
Non-parametric estimators of the stable tail dependence function (STDF): \hat{l}_k(x) and \tilde{l}_k(x).
stdf(x, k, X, alpha = 0.5) stdf2(x, k, X)
x |
A d-dimensional point to estimate the STDF in. |
k |
Value of the tail index k. |
X |
A data matrix of dimensions n by d with observations in the rows. |
alpha |
The parameter α of the estimator \hat{l}_k(x) ( |
The stable tail dependence function in x can be estimated by
\hat{l}_k(x) = 1/k ∑_{i=1}^n 1_{\{\exists j\in\{1,…, d\}: \hat{F}_j(X_{i,j})>1-k/n x_j\}}
with
\hat{F}_j(X_{i,j})=(R_{i,j}-α)/n
where R_{i,j} is the rank of X_{i,j} among the n observations in the j-th dimension:
R_{i,j}=∑_{m=1}^n 1_{\{X_{m,j}≤ X_{i,j}\}}.
This estimator is implemented in stdf.
The second estimator is given by
\tilde{l}_k(x) = 1/k ∑_{i=1}^n 1_{\{X_{i,1}≥ X^{(1)}_{n-[kx_1]+1,n} or … or X_{i,d}≥ X^{(d)}_{n-[kx_d]+1,n}\}}
where X_{i,n}^{(j)} is the i-th smallest observation in the j-th dimension.
This estimator is implemented in stdf2.
See Section 4.5 of Beirlant et al. (2016) for more details.
stdf returns the estimate \hat{l}_k(x) and stdf2 returns the estimate \tilde{l}_k(x).
Tom Reynkens
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.
# Generate data matrix X <- cbind(rpareto(100,2), rpareto(100,3)) # Tail index k <- 20 # Point to evaluate the STDF in x <- c(2,3) # First estimate stdf(x, k, X) # Second estimate stdf2(x, k, X)
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