On some representations of the Bootstrap

Summary The bootstrap statistic is represented as the sum of two terms, one of which has the same distribution as and the other is a relatively negligible remainder. The representation is then applied to the problems of estimating a distribution and a quantile. The study includes the tail part of the distribution and quantiles for which the convergence rates could be faster than the usual n ^-1/2 rate.     

On some finitely based representations of semigroups

In this paper we present a method of obtaining finitely based linear representations of possibly infinitely based semigroups.

     

On some inequalities for the expectation and variance

A variety of bounds are obtained for the variance and expectation of a continuous random variable whose p.d.f. is defined over a finite interval. Previous results are shown to be particular cases of the current more general development.     

On some series representations of the Hurwitz zeta function

A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants.     

On some representations of solutions of dynamical problems in elasticity theory

The divergence and rotation of a translation vector field, which are solutions of wave equations, are used as auxiliary potentials for numerical solution of dynamical problems in elasticity theory. In a particular case, the problems of longitudinal translation are tested in two variants of numerical algorithms with their programming realization, some suggestions are given regarding the choice of the optimal algorithm for achieving highly precise calculations with a small number of discretization points.Translated from Dinamicheskie Sistemy, No. 9, pp. 27–33, 1990.     

Estimating sample mean under interval uncertainty and constraint on sample variance

Traditionally, practitioners start a statistical analysis of a given sample x₁, … , x_n by computing the sample mean E and the sample variance V. The sample values x_i usually come from measurements. Measurements are never absolutely accurate and often, the only information that we have about the corresponding measurement errors are the upper bounds Δ_i on these errors. In such situations, after obtaining the measurement result , the only information that we have about the actual (unknown) value x_i of the ith quantity is that x_i belongs to the interval . Different values x_i from the corresponding intervals lead, in general, to different values of the sample mean and sample variance. It is therefore desirable to find the range of possible values of these characteristics when x_i ∈ x_i.Often, we know that the values x_i cannot differ too much from each other, i.e., we know the upper bound V₀ on the sample variance V : V ? V₀. It is therefore desirable to find the range of E under this constraint. This is the main problem that we solve in this paper.     

Self-dual representations of some dyadic groups

Let F be a non-Archimedean local field of residual characteristic two and let d be an odd positive integer. Let D be a central F-division algebra of dimension d ². Let π be one of: an irreducible smooth representation of D  ^× , an irreducible cuspidal representation of GL _d (F), an irreducible smooth representation of the Weil group of F of dimension d. We show that, in all these cases, if π is self-contragredient then it is defined over \mathbb Q{\mathbb Q} and is orthogonal. We also show that such representations exist.     

Stochastic integral representations,stochastic derivatives and minimal variance hedging

The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.     

On some regression relationships relating sample means and successive maxima

In recent years, a number of papers have appeared in which characterizations of a series of distributions in terms of regression properties of order statistics were obtained. In particular, characterizations of the Student’s t ₂-distribution in terms of linear regression relation-ships, including the sample median X _{2, 3} and the sample mean T ₃, were obtained in 2002. Later, similar results were also established for samples of larger sizes, and characterizations of families of distributions containing the t ₂-distribution were put forward. These characterizations are based, in particular, on linear regression relationships relating sample means T _{2k + 1} and sample medians X _{k + 1, 2k + 1}, k = 1, 2, ...     

An informational analog of the theorem of independence of sample mean and sample variance

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 63–67, 1985.