Volume 59, Issue 6

A definition of concave integral is given for real-valued maps and with respect to Dedekind complete Riesz space-valued “capacities”. Some comparison results with other integrals are given and some convergence theorems are proved.

In a linear model Y ∼ (Xβ, σ 2 I), powers of tests of H0: H′Xβ= 0 are developed following Pearson’s (1900) formulation. The class considered comprises all tests based on linear statistics A′ Y that have expected value 0 under H0. The standard F-statistic, which is in this class, has good power properties, but others may be preferred in some settings.

A theorem on asymptotic linearity of L-estimates is proved under general set of regularity conditions, allowing the sampled distribution to be non-integrable. The main result is the improvement in the order of the remainder term in the formula for asymptotic linearity of L-statistic. It is shown that in the case of the integral coefficients this term R n = O P(1/n) and the case of functional coefficients is also covered.

Seemingly, testing for fixed effects in linear models with variance-covariance components has been solved for decades. However, even in simple situations such as in fixed one-way model with heteroscedastic variances (a multiple means case of the Behrens-Fisher problem) the questions of statistical properties of various approximations of test statistics are still alive. Here we present a brief overview of several approaches suggested in the literature as well as those available in statistical software, accompanied by a simulation study in which the accuracy of p-values is studied. Our interest is limited here to the Welch’s test, the Satterthwaite-Fai-Cornelius test, the Kenward-Roger test, the simple ANOVA F-test, and the parametric bootstrap test. We conclude that for small sample sizes, regardless the number of compared means and the heterogeneity of variance, the ANOVA F-test p-value performs the best. For higher sample sizes (at least 5 per group), the parametric bootstrap performs well, and the Kenward-Roger test also performs well.

In the paper we solve the problem of D ℋ-optimal design on a discrete experimental domain, which is formally equivalent to maximizing determinant on the convex hull of a finite set of positive semidefinite matrices. The problem of D ℋ-optimality covers many special design settings, e.g., the D-optimal experimental design for multivariate regression models. For D ℋ-optimal designs we prove several theorems generalizing known properties of standard D-optimality. Moreover, we show that D ℋ-optimal designs can be numerically computed using a multiplicative algorithm, for which we give a proof of convergence. We illustrate the results on the problem of D-optimal augmentation of independent regression trials for the quadratic model on a rectangular grid of points in the plane.

Pseudo-rank functions on bounded lattices are introduced and their properties are studied. It is shown that if the set of all pseudo-rank functions on a Boolean lattice is nonempty then it is a Choquet simplex.

In this paper we apply the concept of interval-valued bifuzzy sets to hemirings. We introduce the notion of interval-valued bifuzzy left h-ideals of hemirings and investigate some of their properties. Using interval-valued bifuzzy left h-ideals, characterizations of Artinian and Noetherian hemirings are established.

For a given continuous function F on a compact interval E in the set ℝ of reals the problem is how to describe the “total change” of F on a set M ⊂ E. Full variational measures W F(M) and V F(M) (see Section 2) in the sense presented by B. S. Thomson are introduced in this work to this aim. They are generated by two slightly different interval functions, namely the oscillation of F over an interval and the value of the additive interval function generated by F, respectively. They coincide with the concept of classical total variation if M is an interval and they are zero if on the set M the function F is of negligible variation. The Kurzweil-Henstock integration is shortly described and some of its properties are studied using the variational measure W F(M) for the indefinite integral F of an integrable function f.

The existence of at least one solution of the following multi-point boundary value problem $$ \left\{ \begin{gathered} [\varphi (x'(t))]' = f(t,x(t),x'(t)),t \in (0,1), \hfill \\ x(0) - \sum\limits_{i = 1}^m {\alpha _i x'(\xi _i ) = 0,} \hfill \\ x'(1) - \sum\limits_{i = 1}^m {\beta _i x(\xi _i ) = 0} \hfill \\ \end{gathered} \right. $$ is studied. where α i ∈ ℝ, β i ∈ ℝ for all i = 1, ..., m. Examples are presented to illustrate the main result.

In this article we introduce some vector valued double sequence spaces defined by Orlicz function. We study some of their properties like solidness, symmetricity, completeness etc. and prove some inclusion results.