# Updating least squares

The mainstay in our presented algorithm for the solution of LSE problem is the updating procedure.Therefore, our main concern is to study the error analysis of the updating steps.

An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator.The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square.Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods.This includes basis pursuit (BP), basis pursuit denoising (BPDN), and NNLS (non-negative least squares).For certain matching values of $$\lambda$$ and $$\tau$$, BPDN is equivalent to the Lasso problem: minimize $$\frac12 \|Ax-b\|_2^2$$ subject to $$\|x\|_1 \le \tau$$. For singular systems, computes the minimum-norm solution.) LSQR: MATLAB, Fortran, C, C , .

(Iterative method; more stable than symmetric conjugate-gradient method on normal equations.

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.

We reduce the constrained problem to unconstrained linear least squares and partition it into a small subproblem. We carry out the error analysis of the proposed algorithm to show that it is backward stable.

We also illustrate the implementation and accuracy of the proposed algorithm by providing some numerical experiments with particular emphasis on dense problems. It arises in important applications of science and engineering such as in beam-forming in signal processing, curve fitting, solutions of inequality constrained least squares problems, penalty function methods in nonlinear optimization, electromagnetic data processing and in the analysis of large scale structure [) can be obtained using direct elimination, the nullspace method and method of weighting.

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