ReIns (version 1.0.10)

stdf: Non-parametric estimators of the STDF

Description

Non-parametric estimators of the stable tail dependence function (STDF): \(\hat{l}_k(x)\) and \(\tilde{l}_k(x)\).

Usage

stdf(x, k, X, alpha = 0.5)
stdf2(x, k, X)

Value

stdf returns the estimate \(\hat{l}_k(x)\) and stdf2 returns the estimate \(\tilde{l}_k(x)\).

Arguments

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 \(\alpha\) of the estimator \(\hat{l}_k(x)\) (stdf), default is 0.5. This argument is not used in stdf2.

Author

Tom Reynkens

Details

The stable tail dependence function in \(x\) can be estimated by $$ \hat{l}_k(x) = 1/k \sum_{i=1}^n 1_{\{\exists j\in\{1,\ldots, d\}: \hat{F}_j(X_{i,j})>1-k/n x_j\}}$$ with $$\hat{F}_j(X_{i,j})=(R_{i,j}-\alpha)/n$$ where \(R_{i,j}\) is the rank of \(X_{i,j}\) among the \(n\) observations in the \(j\)-th dimension: $$R_{i,j}=\sum_{m=1}^n 1_{\{X_{m,j}\le X_{i,j}\}}.$$ This estimator is implemented in stdf.

The second estimator is given by $$ \tilde{l}_k(x) = 1/k \sum_{i=1}^n 1_{\{X_{i,1}\ge X^{(1)}_{n-[kx_1]+1,n} or \ldots or X_{i,d}\ge X^{(d)}_{n-[kx_d]+1,n}\}}$$ 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.

References

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.

Examples

# 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)