stdf {ReIns}R Documentation

Non-parametric estimators of the STDF

Description

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

Usage

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

stdf2(x, k, X)

Arguments

x

A dd-dimensional point to estimate the STDF in.

k

Value of the tail index kk.

X

A data matrix of dimensions nn by dd with observations in the rows.

alpha

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

Details

The stable tail dependence function in xx can be estimated by

l^k(x)=1/ki=1n1{j{1,,d}:F^j(Xi,j)>1k/nxj} \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

F^j(Xi,j)=(Ri,jα)/n\hat{F}_j(X_{i,j})=(R_{i,j}-\alpha)/n

where Ri,jR_{i,j} is the rank of Xi,jX_{i,j} among the nn observations in the jj-th dimension:

Ri,j=m=1n1{Xm,jXi,j}.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

l~k(x)=1/ki=1n1{Xi,1Xn[kx1]+1,n(1)ororXi,dXn[kxd]+1,n(d)} \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 Xi,n(j)X_{i,n}^{(j)} is the ii-th smallest observation in the jj-th dimension. This estimator is implemented in stdf2.

See Section 4.5 of Beirlant et al. (2016) for more details.

Value

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

Author(s)

Tom Reynkens

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)

[Package ReIns version 1.0.14 Index]