This paper proposes a test for cross-sectional independence with high dimensional panel data. It uses the random matrix theory based approach of Srivastava (2005) in the presence of a large number of cross-sectional units and time series observations. Because the errors are unobservable, the residuals from the regression model for panel data are used. We develop a bias-corrected test after adjusting for the contribution from the regressors. With the aid of the martingale central limit theorem, we prove that the limiting null distribution of the proposed test statistic is normal under mild conditions as cross-sectional dimension and time dimension go to infinity together. We further study the asymptotic relative efficiency of our proposed test with respect to the state-of-art Lagrange multiplier test. An interesting finding is that the newly proposed test can have substantial power gain when the underlying variance magnitudes are not identical across different units.

Tweedie exponential dispersion (TED) process is a large class of generalized stochastic process, which includes the commonly used Wiener process, Gamma process and inverse Gausssian process as special cases. In this paper, we propose an accelerated degradation model based on the TED process, and then apply the objective Bayesian method to make inferences on the model. The Jeffreys prior and reference priors under different group orderings are derived, and the posterior property based on each prior is then validated. A simulation study is carried out to assess the performance of the proposed Bayesian method in comparison with the maximum-likelihood estimation (MLE), and finally the method is applied to analyse a real degradation dataset.

In this paper, for the problem of heteroskedastic general linear hypothesis testing (GLHT) in high-dimensional settings, we propose a random integration method based on the reference L2-norm to deal with such problems. The asymptotic properties of the test statistic can be obtained under the null hypothesis when the relationship between data dimensions and sample size is not specified. The results show that it is more advisable to approximate the null distribution of the test using the distribution of the chi-square type mixture, and it is shown through some numerical simulations and real data analysis that our proposed test is powerful.

In this paper, we propose a new scale-invariant test for linear hypothesis of mean vectors with heteroscedasticity in high-dimensional settings. Most existing tests impose strong conditions on covariance matrices so that null distributions of their tests are asymptotically normal, which restricts the application of test procedures. However, our proposed test has different null distributions under mild conditions. Additionally, the well-known Welch-Satterthwaite chi-square approximation we adopted can automatically mimic the shapes of the null distributions of the test statistic. The performances of the test are illustrated by simulation and real data in finite samples which show that it has robustness and is more powerful than three competitors.

This paper is devoted to the study of the general linear hypothesis testing (GLHT) problem of multi-sample high-dimensional mean vectors. For the GLHT problem, we introduce a test statistic based on L^2-norm and random integration method, and deduce the asymptotic distribution of the statistic under given conditions. Finally, the potential advantages of our test statistics are verified by numerical simulation studies and examples.

In this paper, we propose a new test for linear hypothesis of k-sample mean vectors in high-dimensional normal models based on generalized likelihood ratio method. The proposed test is designed for the "large p small n" situation where the data dimension p is much larger than the sample size n. The asymptotic null and non null distributions of the proposed test are derived under mild conditions. Simulation results show that our new test outperforms some competitors in both size and power. Moreover, our new test can also be applied to non normal data.

1 引言 题目 (2005 年罗马尼亚数学奥林匹克试题)已知a,b,c>0,证明:a+b/c2 +b+c/a2 +c+a/b2 ≥2( 1/a+ 1/b + 1/c )(1). 文[1]给出了其一个命题及推论,下面利用切比雪夫不等式进行统一推广,并给出该试题的另一个推广.

1 引言 生活中的竞技比赛,无论遵循什么样的比赛规则,首先要满足公平性.在保证公平的条件下,为了使比赛场面更加激烈,选择更为适合的比赛规则,使水平较高的参赛者也不能确保获胜,从而形成更加激烈的比赛场面,有助于提高比赛的观赏性和趣味性,推动比赛项目进一步的发展.许多学者对此进行了研究,向明华通过概率的可列可加性将建立的概率差分方程化为等价且易解的代数方程,得到了两人对弈时在k 局连胜规则下的获胜概率[1].

In this paper, the problem of testing the equality of the mean vectors of k populations with possibly unknown and unequal covariance matrices is investigated in high-dimensional settings. The null distributions of most existing tests are asymptotically normal which inevitably imposes strong conditions on covariance matrices. However, we assume here only mild additional conditions on the proposed test, which offers much flexibility in practical applications. Additionally, the Welch-Satterthwaite chi 2-approximation we adopted can automatically mimic the shape of the null distribution of the proposed test statistic, while the normal approximation cannot achieve the adaptivity. Finally, an extensive simulation study shows that the proposed test has better performance on both size and power compared with existing methods.

新时代我国经济已经由高速增长阶段转向高质量发展阶段.针对中部地区 6 个省份的面板数据,在推动中部地区高质量发展的背景下构建了全新的经济发展质量的指标评价体系.在此基础上对中部地区的经济发展质量进行了指标分析、省份分析、类型划分以及时空演变分析,旨在从总体和部分,时间和空间两角度深入探究中部地区的经济发展趋势.研究发现:4 个一级指标中开放共享所占权重最大,这说明当前开放性与共享性已经成为中部地区经济发展质量的首要影响因素;2013-2019 年中部地区经济发展质量的空间聚集分布呈现正分布,总体经济发展质量水平明显提升,其中江西、山西、湖南三个省份的经济发展质量呈现较快提升的趋势;经济发展质量的指数增长值、指数增速以及综合指数值的排名并不一致.基于以上研究结果,提出山西、江西相对而言经济发展任务是比较艰巨的;中部地区各省份之间可以积极主动扩大贸易往来,利用各个地区的特殊条件实现优势互补和互利共赢;在对一个地区进行经济质量衡量的时候,应当多方位、多角度进行考量,避免主观因素的影响.

In this paper, the problem of testing the hypothesis of linear combination of k-sample means of high-dimensional data is investigated under a low-dimensional factor model. We propose a new test and derive that the asymptotic distribution of the test statistic is a weighted distribution of independent chi-squared distribution of 1 degree of freedom under the null hypothesis and mild conditions. We provide numerical studies on both sizes and powers to illustrate performance of the proposed test.

In this paper, a new test statistic based on the weighted Frobenius norm of covariance matrices is proposed to test the homogeneity of multi-group population covariance matrices. The asymptotic distributions of the proposed test under the null and the alternative hypotheses are derived, respectively. Simulation results show that the proposed test procedure tends to outperform some existing test procedures.

In this paper, the problem of high-dimensional multivariate analysis of variance is investigated under a low-dimensional factor structure which violates some vital assumptions on covariance matrix in some existing literature. We propose a new test and derive that the asymptotic distribution of the test statistic is a weighted distribution of chi-squares of 1 degree of freedom under the null hypothesis and mild conditions. We provide numerical studies on both sizes and powers to illustrate performance of the proposed test.

In order to investigate linearly admissible estimators of the common mean parameter in general linear models, we introduce and motivate the use of a balanced loss function obtained by combining Zellner’s idea of balanced loss (Zellner, 1994) with the unified theory of least squares (Rao, 1973). In classes of homogeneous and non-homogeneous linear estimators, sufficient and necessary conditions for linear estimators of the common mean parameter to be admissible are obtained, respectively. A comparison is then made between linearly admissible estimators and a “truly” unified least square estimator.

In this paper, the problem of simultaneously testing mean vector and covariance matrix of one-sample population is investigated in high-dimensional settings. We propose a new test statistic and obtain its asymptotic distributions under null and local alternative hypotheses, respectively. Our asymptotic result for proposed test does not need some conditions such as linearity between the sample size and dimension used in existing studies. Simulation results also demonstrate our new test not only can control reasonably the nominal level but also has greater empirical powers than competing tests.

In the high dimensional setting, this article explores the problem of testing the complete independence of random variables having a multivariate normal distribution. A natural high-dimensional extension of the test in Schott (2005) is proposed for this purpose. The newly defined tests are asymptotically distribution-free as both the sample size and the number of variables go to infinity and hence have well-known critical values, accommodate situations where the number of variables is not small relative to the sample size and are applicable without specifying an explicit relationship between the number of variables and the sample size. In practice, as the true alternative hypothesis is unknown, it is unclear how to choose a powerful test. For this, we further propose an adaptive test that maintains high power across a wide range of situations. An extensive simulation study shows that the newly proposed tests are comparable to, and in many cases more powerful than, existing tests currently in the literature.

We use distance covariance to introduce novel consistent tests of heteroscedasticity for nonlinear regression models in multidimensional spaces. The proposed tests require no user-defined regularization, which are simple to implement based on only pairwise distances between points in the sample and are applicable even if we have non-normal errors and many covariates in the regression model. We establish the asymptotic distributions of the proposed test statistics under the null and alternative hypotheses and a sequence of local alternatives converging to the null at the fastest possible parametric rate. In particular, we focus on whether and how the estimation of the finite-dimensional unknown parameter vector in regression functions will affect the distribution theory. It turns out that the asymptotic null distributions of the suggested test statistics depend on the data generating process, and then a bootstrap scheme and its validity are considered. Simulation studies demonstrate the versatility of our tests in comparison with the score test, the Cramér-von Mises test, the Kolmogorov-Smirnov test and the Zheng-type test. We also use the ultrasonic reference block data set from National Institute for Standards and Technology of USA to illustrate the practicability of our proposals.

The accelerated degradation test (ADT) is an effective method for evaluating the lifetime of high-reliability products. In this paper, a doubly accelerated degradation model based on the inverse Gaussian process is proposed to characterize the ADT data, and then an objective Bayesian approach is presented to analyze the model. Some important noninformative priors including the Jeffreys prior and reference priors under different group orderings are derived. The propriety of the posterior distributions under each prior is validated. A simulation study is carried out to show the superiority of objective Bayesian approach compared with the parametric Bootstrap method. Finally, the approach is applied to analyze a carbon film data.

以安徽省为例,对影响农户固定资产投资行为的因素——耕地面积、平均每个劳动力负担人口、工资性收入、城乡消费水平对比及家庭生产经营支出建立了VAR模型和改进的双对数模型,并根据模型结果,深入探讨如何提高农户的投资水平.

In this paper, we propose a new scalar-transformation-invariant test for linear hypothesis on mean vectors of normal population with unequal covariance matrices in high-dimensional data. The asymptotic null and non-null distributions of our new test are obtained under some regularity conditions. The performance of the proposed test is conducted by numerical simulation and a real data example, which illustrates our new test outperforms competitors in the considered cases. Moreover, numerical studies show that our new test can also be applied to non-normal data.