Теория вероятностей и математическая статистика в задачах
This article concerns the problem of predicting the size of company's customer base in case of solving the task of managing its clients. The author purposes a new approach to segment-oriented predicting the size of clients based on adopting the Staroverov's employees moving model. Besides the article includes the limitations of using this model and its modification for each type of relations of the client and the company.
This paper examines two Markov chain Monte Carlo methods that have been widely used in econometrics. An introductory exposition of the Metropolis algorithm and the Gibbs sampler is provided. These methods are used to simulate multivariate distributions. Many problems in Bayesian statistics can be solved by simulating the posterior distribution. Invariance condition is of importance, the proofs are given for both methods. We use finite Markov chains to explore and substantiate the methods. Several examples are provided to illustrate the applicability and efficiency of the Markov chain Monte Carlo methods. They include bivariate normal distribution with high correlation, bivariate exponential distribution, mixture of bivariate normals.
We develop a consensus clustering framework developed three decades ago in Russia and experimentally demonstrate that our least squares consensus clustering algorithm consistently outperforms several recent consensus clustering methods.
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.