A note on the bootstrap method for testing the existence of finite moments
This paper discusses a bootstrap-based test, which checks if finite moments exist, and indicates cases of possible misapplication. It notes, that a procedure for finding the smallest power to which observations need to be raised, such that the test rejects a hypothesis that the corresponding moment is finite, works poorly as an estimator of the tail index or moment estimator. This is the case especially for very low- and high-order moments. Several examples of correct usage of the test are also shown. The main result is derived analytically, and a Monte-Carlo experiment is presented.
A computable estimate of the readiness coefficient for a standard binary-state system is established in the case where both working and repair time distributions possess heavy tails.
We present a new method for generating pseudo random sequences which is based on represenation of real numbers in some numeral systems. We describe an algorithm for generation sequences, present experimental results and methodology of randomness checking.
We consider a transformed Ornstein--Uhlenbeck process model that can be a good candidate for modeling real-life processes characterized by a combination of time-reverting behavior with heavy distribution tails. We begin with presenting the results of an exploratory statistical analysis of the log prices of a major Australian public company, demonstrating several key features typical of such time series. Motivated by these findings, we suggest a simple transformed Ornstein--Uhlenbeck process model and analyze its properties showing that the model is capable of replicating our empirical findings. We also discuss three different estimators for the drift coefficient in the underlying (unobservable) Ornstein--Uhlenbeck process which is the key descriptor of dependence in the process.
Ensemble summary statistics represent multiple objects on the high level of abstraction—that is, without representing individual features and ignoring spatial organization. This makes them especially useful for the rapid visual categorization of multiple objects of different types that are intermixed in space. Rapid categorization implies our ability to judge at one brief glance whether all visible objects represent different types or just variants of one type. A framework presented here states that processes resembling statistical tests can underlie that categorization. At an early stage (primary categorization), when independent ensemble properties are distributed along a single sensory dimension, the shape of that distribution is tested in order to establish whether all features can be represented by a single or multiple peaks. When primary categories are separated, the visual system either reiterates the shape test to recognize subcategories (indepth processing) or implements mean comparison tests to match several primary categories along a new dimension. Rapid categorization is not free from processing limitations; the role of selective attention in categorization is discussed in light of these limitations.
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.