Topological regular variation: II. The fundamental theorems

This paper investigates fundamental theorems of regular variation (Uniform Convergence, Representation, and Characterization Theorems) some of which, in the classical setting of regular variation in R, rely in an essential way on the additive semigroup of natural numbers N (e.g. de Bruijn's Representation Theorem for regularly varying functions). Other such results include Goldie's direct proof of the Uniform Convergence Theorem and Seneta's version of Kendall's theorem connecting sequential definitions of regular variation with their continuous counterparts (for which see Bingham and Ostaszewski (2010) [13]). We show how to interpret these in the topological group setting established in Bingham and Ostaszewski (2010) [12] as connecting N-flow and R-flow versions of regular variation, and in so doing generalize these theorems to Rd. We also prove a flow version of the classical Characterization Theorem of regular variation.     

Tight frames of compactly supported multivariate multi-wavelets

This paper is devoted to the study and construction of compactly supported tight frames of multivariate multi-wavelets. In particular, a necessary condition for their existence is derived to provide some useful guide for constructing such MRA tight frames, by reducing the factorization task of the associated polyphase matrix-valued Laurent polynomial to that of certain scalar-valued non-negative ones. We illustrate our construction method with examples of both multivariate scalar- and vector-valued subdivision schemes. Since our constructions for C¹ and C² piecewise cubic schemes are quite involved, we also include the corresponding Matlab code in the Appendix.     

限制参数空间双侧检验P值的修正及与贝叶斯证据的调停一致

本文研究了在限制参数空间双侧检验的P值的修正及基于修正P值的贝叶斯检验和经典统计检验的调停.研究表明Wang(2006)给出的修正P值存在重要缺陷,并给出了一种新的修正P值,该修正P值具有较合理的性质,并由此可一定程度缓和调停贝叶斯检验和经典统计检验的存在的冲突.     

A generalized Goulden–Jackson cluster method and lattice path enumeration

The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths – both with height bounded and unbounded – by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulas.     

Analysis of reliability systems via Gini-type index

Different strategies of reliability theory for the analysis of coherent systems have been studied by various researchers. Here, the Gini-type index is utilized as an applicable tool for the study and comparison of the ageing properties of complex systems. A new stochastic order in terms of Gini-type index is introduced to compare the speed of ageing of components and systems. The parallel-series and series-parallel systems with shared components are studied by their corresponding Gini-type indexes. Also, the generalization of Gini-type index for the multidimensional case is discussed, and is used to compare components lifetimes properties in the presence of other dependent components. It is shown that the ageing properties of a component lifetime can differ when the other components are working or have already failed. Numerous illustrative examples are given for better intuition of Gini-type and generalized Gini-type indexes throughout the paper.     

Hájek-Šidák approach to the asymptotic distribution of multivariate rank order statistics

Let X_α = (X_1α,…, X_pα), 1 ≤ α ≤ N_ν, ν ≥ 1 be N_ν independent observations from a density function f(x) where x ∈ R^p, the p-dimensional real space. Let R_νjα denote the rank of X_jα in the ordered array of X_j1 ,…, X_jNν; 1 ≤ j ≤ p and consider the multivariate rank order statistics

$\text{T}{}_{\mathrm{vj}}\text{=}\text{∑}\text{α = 1}\text{N}{}_{\mathrm{v}}\text{C}{}_{\mathrm{y\alpha }}\text{a}{}_{\mathrm{vj}}\text{(R}{}_{\mathrm{vj\alpha }}\text{),}$

where the constants, c_να, 1 ≤ α ≤ N_ν satisfy the Noether condition and the scores, a_νj(α), 1 ≤ j ≤ p, 1 ≤ α ≤ N_ν converge as ν → ∞, for each j, in quadratic mean to a nonconstant, square integrable function π_j(u), 0 < u < 1. Under the hypothesis of randomness, the joint asymptotic conditional and uncoditional normality of the statistics T_νj, 1 ≤ j ≤ p is established. Further, under mild conditions on the underlying density functions and assuming contiguous location shift alternatives, the joint asymptotic normality of these statistics is also established.     

Measurement uncertainty in water microbiology may not estimate the laboratory’s precision

It has been proposed that microbiology laboratories introduce a measurement uncertainty for organism counts. Suggested methods of achieving this involve studying replicate counts from subsamples. This will overwhelmingly estimate the natural variation in numbers of organisms present in different subsamples, and the different characteristics of strains of organisms within the species or group (e.g. the coliform bacteria) being counted. Thus any measure of the laboratorys competence, in terms of precision, will be masked. In chemistry there is likely to be much more complete homogeneity of an analyte in a test material and these replicate measures will illustrate whether the laboratory is competent in achieving accurate results.     

Strong tractability of integration using scrambled Niederreiter points

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The -exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

     

Modelling of extreme wave heights and periods through copulas

Optimal design of coastal or offshore structures requires the estimation of extreme quantiles of oceanographic data such as wave heights and wave periods. Since there are strong correlations between oceanographic variables, it is necessary to use multivariate models in order to capture its dependencies. To achieve this, an approach based on copulas is proposed and is compared to a model based on the physical behaviour of waves.