关于高维统计模型的似然比检验

发布时间：2018-12-05 19:53

【摘要】：在大数据时代,我们经常会遇到很多高维数据方面的问题,这些问题一般都具有维数p和样本容量n都很大的特征,通常也被称为“大p、大n”问题.传统的多元统计分析可以很好地解决维数p很小或者是固定情况下的问题,例如经典的卡方逼近方法,似然比检验方法等,但随着维数p的增加,这些方法不能很好地解决问题甚至失效.因此,寻找一些能够解决高维问题的新方法是非常有意义的.本文主要考虑了维数p和样本容量n都很大的两个模型的高维假设检验问题.首先考虑的是具有循环对称协方差结构的高维假设检验.在两种稍有不同的假设下,利用矩母函数的连续性定理,通过伽马函数的渐近展开,证明了在正态总体下,当原假设成立时,似然比统计量依分布收敛于一个正态分布的随机变量,然后将本文提出的的高维似然比检验方法(HLRT)同卡方逼近方法(BOX)、高维edgeworth展开方法(HEE)以及更精确的高维edgeworth展开方法(AHEE)进行数据模拟,结果表明本文提出的HLRT方法优于BOX方法和HEE方法,并且和AHEE方法在处理高维数据方面一样好.第三章研究高维主成分分析中最小特征值等价性的似然比检验.在原假设以及正态总体的假定下,利用特征函数的连续性定理和类似的展开方法,得到了对数形式的似然比统计量服从正态分布.数值模拟揭示了本文提出的正态逼近方法(HLRT)和更精确的高维渐近展开方法(AHAE) 一样好,并且在维数p增加的过程中,它们两个方法都比卡方逼近方法(Lawley)的结果要准确.
[Abstract]:In big data's time, we often encounter a lot of problems in high-dimensional data. These problems are usually characterized by large dimension p and sample size n, which are usually called "big p, large n" problems. Traditional multivariate statistical analysis can solve the problem of small dimension p or fixed dimension, such as classical chi-square approximation method, likelihood ratio test method, etc. These methods do not solve problems or even fail. Therefore, it is very meaningful to find some new methods to solve high dimensional problems. In this paper, we consider the problem of high dimensional hypothesis testing for two models with large dimension p and sample size n. The first consideration is the test of high dimensional hypothesis with cyclic symmetric covariance structure. Under two slightly different assumptions, by using the continuity theorem of moment generating function and the asymptotic expansion of gamma function, it is proved that in normal population, when the original hypothesis holds, The likelihood ratio statistic converges to a random variable of normal distribution according to the distribution. Then, the high dimensional likelihood ratio test method (HLRT) and chi-square approximation method (BOX), are proposed in this paper. The high dimensional edgeworth expansion method (HEE) and the more accurate high dimensional edgeworth expansion method (AHEE) are simulated. The results show that the proposed HLRT method is better than the BOX method and the HEE method, and is as good as the AHEE method in dealing with high-dimensional data. In chapter 3, the likelihood ratio test of minimum eigenvalue equivalence in high dimensional principal component analysis is studied. Under the original assumption and the assumption of normal population, by using the continuity theorem of the characteristic function and the similar expansion method, the logarithmic form of likelihood ratio statistics is obtained from the normal distribution. The numerical simulation shows that the normal approximation method (HLRT) proposed in this paper is as good as the more accurate high dimensional asymptotic expansion method (AHAE), and in the process of increasing the dimension p, Both of them are more accurate than the chi-square approximation method (Lawley).
【学位授予单位】：河南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1

【相似文献】