Vol 37, No 4 (2016)

Extreme positive dependence, also known as comonotonicity or the Fréchet upper bound, has been an important concept in insurance since it describes the most dangerous financial behaviors. However, there is no generally agreed upon definition for extreme negative dependence because the corresponding Fréchet lower bound does not exist. To resolve this, a set of copulas under the name of d-Countermonotonicity (d-CTM) has been proposed to define extreme negative dependence rather than a single copula. The set of d-CTM copulas can be quite useful in various optimization problems. In this paper, we investigate various properties of d-CTM with more emphasis on the practical issues of actual optimization problems. As an application to insurance, we explore the effect of risk pooling under extreme dependence by adopting the so-called measure of uncertainty.

Under the conditions of integrability the conditional version E(f(X)G)E(g(X)|G) ≤ E(f(X)g(X)|G) a.s. of Chebyshev’s other inequality is proved for monotonic functions f and g of the samemonotonicity, for any random variable X, and for any σ-algebra G. An improved conditional version of the Grüss inequality is also proved.

The goal of this article is to show, that quality control based on extremal observation values is inconsistent in situations, when quality of statistical procedure is the average error rate of those experiments which ended with the adoption of some decision. Thanks to L.N. Bolshev this concept of risk procedures was introduced in the 70s–80s of the last century (see [1, 2], where such risk was called d-posteriori). In this article we analyze two probabilistic models, in which distributions of observations and output parameter are exponential. Using the asymptotic analysis methods of Laplace integrals we can show, that values of d-posteriori probabilistic errors do not reach arbitrary limitation for the procedure based on the first (last) order statistics.

The ultrapower of real line, R_U, where U is a nontrivial ultrafilter in the set the N of natural integers, is some realizations of the “non-standard expansion” *R of the set of real numbers. Due to “good” properties of the factorization of cartesian productwith respect to ultrafilter, ultraproducts hold a number of considerable value properties from the algebraic point of view. At the same time it is not any good “natural” (i.e. determined by the topology of factors) topology. In this article some properties of the Gaussian measure defined on ultraproduct of linear measurable spaces are investigated. In particular, we will give an example of a Gaussian not extreme measure. It will be defined on the linear measurable space which doesn’t have any topological structure. For the proof of many statements of the work the technics of the ultraproducts developed in work [1] is used.

Gakhov class G is formed by the holomorphic and locally univalent functions in the unit disk with no more than unique critical point of the conformal radius. Let D be the classical Dirichlet space, and let P: f ↦ F = f″/f′. We prove that the radius of the maximal ball in P(G)∩D with the center at F = 0 is equal to 2.

A general problem of the interval estimation for a ratio of two proportions p₁/p₂ according to data from two independent samples is considered. Each sample may be obtained in the framework of direct or inverse binomial sampling. Asymptotic confidence intervals are constructed in accordance with different types of sampling schemes with an application, where it is possible, of unbiased estimations of success probabilities and also their logarithms. Since methods of constructing confidence intervals in the situations when values for the both samples are obtained for identical sample schemes (for only direct or only inverse binomial sampling) are already developed and well known, the main purpose of this paper is the investigation of constructing confidence intervals in two cases that correspond to different sampling schemes (one is direct, another is inverse). In this situation it is possible to plan the sample size for the second sample according to the number of successes in the first sample. This, as it is shown by the results of statistical modeling, provides the intervals with confidence level which closer to the nominal value. Our goal is to show that the normal approximations (which are relatively simple) for estimates of p₁/p₂ and their logarithms are reliable for a construction of confidence intervals. The main criterion of our judgment is the closeness of the confidence coefficient to the nominal confidence level. It is proved theoretically and shown by statistically modeled data that the scheme of inverse binomial sampling with planning of the size in the second sample is preferred. Main probability characteristics of intervals corresponding to all possible combinations of sampling schemes are investigated by the Monte-Carlo method. Estimations of coverage probability, expectation and standard deviation of interval widths are collected in tables and some recommendations for an application of each of the intervals obtained are presented. Finally, a sufficient and complete review of the literature for the problem is also presented.

In this paper we consider a sequential d-guaranteed “first crossing” test for distinguishing hypotheses H₀: θ∈ Θ₀ = [a, b] under alternative H₁: θ∈ Θ₀ for the mean of a normal distribution N(θ, σ²). Unknown value θ is a realization of the random variable ϑ with a prior normal distribution N(μ, τ²). The continuation region of the experiment is written implicitly and also its graphical illustration is given. We suggest a modification of this region, which understate guarantee of the sequential test, but has an explicit form. Also, our goal is to explore the characteristics of the moment of stopping of the statistical experiment(sample size) using statistical modelingmethod and to illustrate matching the characteristics of a modified test to nominal values of d-risk.

This paper deals with problem of distinguishing between the two hypotheses H₀: θ ≤ 0, H₁: θ > 0 based on a fixed volume sample with a normal distribution N(θ, 1), θ ∈ R. It is considered by suppose that the true value θ is a realization of a random value ϑ with some unknown a priori density g(θ). An empirical estimate g(θ) based on the estimate of archive data of prior distribution characteristic function is suggested for the d-risk of the optimal criteria (conditional probability of justice of hypothesis H_j in condition that it is rejected, j = 0, 1). Consistency of empirical estimators of d-risks and appropriate critical values are studied. The rate of convergence is discovered from obtained estimates.