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## Comparison of efficiency of estimates by the methods of least absolute deviations and least squares in the autoregression model with random coefficient

For the model of autoregression with a random coefficient, the estimate by the least absolute deviations (LAD) method was proved to be consistent and asymptotically normal. For the asymptotic relative efficiency of the estimate by the LAD method as compared to the least squares method, an analytical expression was obtained. For the case where the innovative field of the autoregression process has the Tukey distribution, consideration was given to the behavior of the relative asymptotic efficiency.

We consider LS-, LAD-, R-, M-, S-, LMS-, LTS-, MM-, and HBR-estimates for the parameters of a linear regression model with unknown noise distribution. With computer modeling for medium sized samples, we compare the accuracy of the considered estimates for the most popular probability distributions of noise in a regression model. For different noise distributions, we analytically compute asymptotic efficiencies of LS-, LAD-, R-, M-, S-, and LTS- estimates. We give recommendations for practical applications of these methods for different noise distributions in the model. We show examples on real datasets that support the advantages of robust estimates.

Using computer simulation and a study of the asymptotic distribution, we consider the relative efficiency of M-estimates for the coefficients of the threshold autoregressive equation with respect to the least squares and least absolute deviation estimates. We assume that the updating sequence of the autoregressive equation can have Student’s, logistic, double exponential, normal, or contaminated normal distributions. We prove asymptotic normality of M-estimates with a convex loss function.

The paper covers mathematical and heuristic approaches for solution the image restoration problem. Attention is paid to the least squares method, least absolute deviations, Tikhonov regularization, total variation, Wiener and Kalman filters, as well as matched filter. A description of a new method for constructing the maximum likelihood estimate is given. Such heuristic approaches as the non-local means, block-matching and 3D filtering, K-SVD were also considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.