Variance Reduction in Monte Carlo Estimators via Empirical Variance Minimization
For Monte Carlo estimators, a variance reduction method based on empirical variance minimization in the class of functions with zero mean (control functions) is described. An upper bound for the efficiency of the method is obtained in terms of the properties of the functional class.
We evaluate the sporting effects of the seeding system reforms in the Champions League, the major football club tournament organized by the Union of European Football Associations (UEFA). Before the 2015-16 season, the teams were seeded in the group stage by their ratings. Starting from the 2015-16 season, national champions of the Top-7 associations are seeded in the first pot, whereas other teams are seeded by their rating as before. Taking effect from the season 2018-19, the team's rating will not longer include 20% of the rating of the association that the team represents. Using the prediction model, we simulate the whole UEFA season and obtain numerical estimates for competitiveness changes in the UEFA tournaments caused by these seeding reforms. We report only marginal changes in tournament metrics that characterize ability of the tournament to select the best teams and competitive balance. Probability of changes in the UEFA national association ranking does not exceed several percent for any association.
The article presents the analysis of the measures of risk non-financial company. Identified key risk metrics. If justified the use of EVaR models. Developed methodical recommendations on the use of EVaR in stress-testing company.
This paper presents a simple bootstrap test to verify the existence of finite moments. The efficacy of the test relies on the fact that in the absence of a first moment and under certain general conditions, the arithmetic average of a sample grows at a rate greater than the growth rates of the arithmetic averages of the sub-samples. Firstly, we show test consistency analytically. Then, Monte-Carlo simulations are performed to compare our test with the Hill estimator.
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.