A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses

We are concerned with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and consider the computation of multiplicity-adjusted p-values under the respective global hypothesis. By means of numerical examples, we demonstrate how much gain in level exhaustion or, equivalently, power can be achieved with corresponding multivariate multiple tests compared with approaches which are only based on univariate marginal distributions and do not take the dependence structure among the test statistics into account. As a further contribution of independent value, we provide an overview of essentially all analytic formulas for computing multivariate chi-square probabilities of the considered types which are available up to present. These formulas were scattered in the previous literature and are presented here in a unified manner.     

Combinatorics and Statistical Issues Related to the Kruskal–Wallis Statistic

We explore criteria that data must meet in order for the Kruskal–Wallis test to reject the null hypothesis by computing the number of unique ranked datasets in the balanced case where each of the m alternatives has n observations. We show that the Kruskal–Wallis test tends to be conservative in rejecting the null hypothesis, and we offer a correction that improves its performance. We then compute the number of possible datasets producing unique rank-sums. The most commonly occurring data lead to an uncommonly small set of possible rank-sums. We extend prior findings about row- and column-ordered data structures.     

An Empirical Analysis of Some Nonparametric Goodness-of-Fit Tests for Censored Data

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ² tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.     

A Nonparametric Test Procedure Based on Quantiles for the Right Censored Data

In this study, we propose nonparametric tests using the several quantile statistics simultaneously for the right censored data. First of all, we consider statistics of the quadratic form with estimated covariance matrices. Then we derive the limiting distribution using the large sample approximation theory. Also we consider different forms of statistics such as the maximal and summing types with their limiting distributions. Then we illustrate our procedure with examples and compare performance among tests with empirical powers through a simulation study. Also we comment briefly on some interesting features including re-sampling methods as concluding remarks. Finally in Appendices, we provide proofs for the theoretic results needed for the derivation of the limiting distributions of the proposed test statistics.     

Static Hedging of Geometric Average Asian Options with Standard Options

In this article, we first establish a theorem that represents the price of an Asian option in terms of standard European options with a shorter term and different strikes. Then using Gauss–Hermite numerical integration, we discretize our theorem so as to use Monte Carlo simulation to examine the error of the static hedging under the Black–Scholes model and the Merton jump-diffusion model. For ease of comparison, we also provide the error of the dynamic hedging. The numerical results show that the static hedging strategy performs better than the dynamic one under both models.     

Mean Time to Failure of Weighted k-out-of-n: G Systems

The purpose of this article is to develop a Monte-Carlo simulation algorithm for computing mean time to failure (MTTF) of weighted-k-out-of-n:G and linear consecutive-weighted-k-out-of-n:G systems. Our algorithm is based on the use of appropriately defined stochastic process which represents the total weight of the system at time t. These stochastic processes are explicitly defined and used along with the ordered component lifetimes to simulate MTTF of the systems with weighted components.     

退溪“礼缘仁情”中的易学思想

退溪学《易》、讲《易》、述《易》完全是为了进修践履。退溪既继承朱子,又与朱子有所不同。他认为“《易》乃理数渊源之书”,而不赞成朱子“《易》本为卜筮之书”的观点。退溪体悟《易》的时中之义至精,且身体力行之。他主张礼有因有革,有常有变,礼在践履中从俗、从宜、从权,其基点是“缘仁情”。     

新冠肺炎疫情影响下的白俄罗斯及其与中国合作

新冠肺炎疫情对白俄罗斯经济、社会和政治造成巨大冲击,暴露并加剧了白俄罗斯国家和社会治理领域存在的问题。在疫情背景下,白俄罗斯第六届总统选举触发政治危机。卢卡申科政府通过采取有效措施应对使紧张局势逐渐趋于缓和,国内恢复正常秩序,内外政策随之调整,对内宣布启动修宪,对外加强非西方外交,尤其加强与俄罗斯的联盟关系。疫情发生以来,中国与白俄罗斯开展了行之有效的合作,对于促进白俄罗斯稳定与发展发挥了积极的作用,并为后疫情时代构建中白卫生健康命运共同体奠定了良好基础。     

中国传统“中庸”思想在音乐发展中的作用

中国传统的"中庸"思想在当代音乐发展的走向中仍具有很好的启示作用。西方文化中的单极思维造成的对"音乐资源"的无限"开采",有可能给音乐的未来发展带来不良后果,使音乐的和声与曲式构架遭到彻底的破坏。中国传统文化中提倡"不偏不倚"的"中庸"思想能避免这两种极端,对音乐的未来发展或许有一些益处。