Lobachevskii Journal of Mathematics Pub Date : 2021-04-18 , DOI: 10.1134/s1995080221020037 F. A. Abdushukurov , A. N. Chuprunov
Abstract
We consider random vectors that consist of numbers of colored balls belonging to a fixed value set in a multi-color urn scheme without return. We proved, that under certain conditions, random vectors, consisting of centered and normed elements of these vectors, converge in distribution to a random vector made up of independent Gaussian random variables with means 0 and variances of 1. We also obtained limit theorems for the functions of these random vectors. Applications of these theorems to estimate probabilities of type I errors of the \(\chi^{2}\)-test and to estimate probabilities of type I errors and type II errors of some statistical tests are given.
中文翻译:
无返回多色Color方案中固定集的彩球数
摘要
我们考虑随机向量,该向量由属于固定值的彩色球数组成,该固定值在多色方案中设置,没有返回值。我们证明了,在某些条件下,由这些向量的中心和范数元素组成的随机向量在分布上收敛到由均值为0且方差为1的独立高斯随机变量组成的随机向量。我们还获得了定理这些随机向量的函数。给出了这些定理在估计\(\ chi ^ {2} \)检验的I型错误的概率以及估算某些统计检验的I型错误和II型错误的概率中的应用。