### Article

## Goodness-of-Fit Tests Based on a Characterization of Logistic Distribution

The logistic family of distributions belongs to the class of important families in the theory of probability and mathematical statistics. However, the goodness-of-fit tests for the composite hypothesis of belonging to the logistic family with unknown location parameter against the general alternatives have not been sufficiently explored. We propose two new goodness-of-fit tests: the integral and the Kolmogorov-type, based on the recent characterization of the logistic family by Hua and Lin. Here we discuss asymptotic properties of new tests and calculate their Bahadur efficiency for common alternatives.

A comparative analysis is presented in the paper of asymptotic efficiencies derived on the basis of the Kaplan-Meier and the maximum likelihood methods which are widely used for censored data problems. A random variable is considered which is distributed according to the truncated normal distribution and is right-censored by another random variable distributed according to one of the following distributions: truncated normal, exponential and uniform one. It is demonstrated that under the correct assumption on the parametric family of distributions the maximum likelihood method yields higher asymptotic efficiency than the Kaplan-Meier method. This advantage of the maximum likelihood method becomes more significant for higher censoring percentage of the observed data sample and for the survival function close to 0 and 1. Besides it is obtained that the relative asymptotic efficiency of both methods under consideration is depended on the type of censored distribution. All the calculations are produced in computer mathematical environment Wolfram *Mathematica.*

The article deals with a model of the dynamics of aggregate banking assets to GDP in the Eurasian Economic Union member states based on the logistic equation. A fuzzy Mamdani model is applied to measure the extent to which this dynamics has an effect on Capital Adequacy Ration as a regulatory standard. Based on this model we consider scenarios of transition to regulatory standards recommended by the Basel Committee on Banking Supervision in the framework of Basel III transformation.

Debt, as one of basic human relations, has profound effects on economic growth. Debt accumulation in the global economy was modeled by the stochastic logistic equation reflecting causality between leverage and its relative rate of change. The model, identifying interactions and feedbacks in aggregate behaviour of creditors and borrowers, addressed various issues of macrofinancial stability. Qualitatively diverse patterns, including the Wicksellian (normal) market, the Minsky financial bubbles and the Fisherian debt-deflation, were discerned by appropriate combinations of rates of return, spreads and leverage. The Kolmogorov-Fokker-Plank equation was used to find out the stationary gamma distribution of leverage that was instrumental for the evaluation of appropriate failure and survival functions. Two patterns corresponding to different forms of a stationary gamma distribution were recognized in the long run leverage dynamics and were simulated as scenarios of a possible system evolution. In particular, empirically parameterized asymptotical distribution indicated excessive leverage and unsustainable global debt accumulation. It underlined the necessity of comprehensive reforms aiming to decrease uncertainty, debt and leverage. Assuming these reforms were successfully implemented, global leverage distributions would have converged in the long run to a peaked gamma distribution with the mode identical to the anchor leverage. The latter corresponded to a balanced long run debt demand and supply, hence to fairly evaluated financial assets fully collateralized by real resources. A particular case of macrofinancial Tobin’s q-coefficients following the Ornstein-Ulenbeck process was studied to evaluate a reasonable range of squeezing the bloated world finance. The model was verified on data published by the IMF in Global Financial Stability Reports for the period 2003-2013.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.