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The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models.In large samples, it is asymptotically normal with an explicit and estimable covariance matrix.Likelihood-based procedures are a common way to estimate tail dependence parameters.They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models.

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(Suitable as basis factorization package for dense simplex method, or for updating sparse factorizations via the Schur-complement (Block-LU) method.) ASP: MATLAB software implementing an active-set method for sparse optimization: minimize \(\lambda \|x\|_1 \frac12 \|Ax-b\|_2^2\).

The presented approach is suitable for dense problems and also applicable where factorization of a problem matrix is available and we are interested in the solution after adding new data to the original problem.

In the future, it will be of interest to study the updating techniques for sparse data problems and for those where the linear least squares problem is fixed and the constraint system is changing frequently.

The estimator is then applied to disentangle sources of tail dependence in European stock markets.

MINRES: Fortran, MATLAB, and Python software for sparse symmetric linear equations \(Ax = b\), where \(A\) is definite or indefinite, possibly singular. Allows a positive-definite preconditioner.) MINRES-QLP: MATLAB software for sparse symmetric linear equations \(Ax = b\) or least-squares problems \( \min \|Ax - b\| \), where \(A\) is definite or indefinite, possibly singular. NET, and Python software for sparse linear equations and sparse least squares.

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