While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few
easily implemented statistics for testing spherical symmetry for the underlying distribution of high-dimensional data. Many
existing statistics for this purpose were constructed by the theory of empirical processes and turn out to converge slowly
to their limiting distributions. Some existing statistics for the same purpose were given in the form of high-dimensional
integrals that are not easily evaluated in numerical computation. In this paper, we develop some necessary tests for spherical
symmetry based on both univariate and multivariate uniform statistics. These statistics are easily evaluated numerically and
have simple limiting distributions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on
controlling type I error rates and power. 相似文献
A new dimension-reduction graphical method for testing high-dimensional normality is developed by using the theory of spherical distributions and the idea of principal component analysis. The dimension reduction is realized by projecting high-dimensional data onto some selected eigenvector directions. The asymptotic statistical independence of the plotting functions on the selected eigenvector directions provides the principle for the new plot. A departure from multivariate normality of the raw data could be captured by at least one plot on the selected eigenvector direction. Acceptance regions associated with the plots are provided to enhance interpretability of the plots. Monte Carlo studies and an illustrative example show that the proposed graphical method has competitive power performance and improves the existing graphical method significantly in testing high-dimensional normality. 相似文献
Organic chemistry is an important basic course for biology-majored students. Starting from the perspective of stimulating students' interest in learning, breaking through the key and difficult points, and realizing the course education, this paper proposes that teachers should combine the characteristics of biology discipline, strengthen the chemical cognition of biologically functional molecules or biomass components, and promote the cultivation of students' interdisciplinary integration consciousness. At the same time, this paper explores the teaching reform of organic chemistry for biology-majored students so as to lay a foundation for cultivating interdisciplinary talents with comprehensive innovation ability. 相似文献
In this paper we employ the conditional probability integral transformation (CPIT) method to transform a d-dimensional sample from two classes of generalized multivariate distributions into a uniform sample in the unit interval \((0,\,1)\) or in the unit hypercube \([0,\,1]^{d-1}\) (\(d\ge 2\)). A class of existing uniform statistics are adopted to test the uniformity of the transformed sample. Monte Carlo studies are carried out to demonstrate the performance of the tests in controlling type I error rates and power against a selected group of alternative distributions. It is concluded that the proposed tests have satisfactory empirical performance and the CPIT method in this paper can serve as a general way to construct goodness-of-fit tests for many generalized multivariate distributions.