多维正态分布均值在序约束下的假设检验

Testing the Multivariate Normal Means Under Order Restriction

在序限制下的统计推断是统计分析中的一个重要领域.保序回归理论在这个领域中起着关键性的作用.多维保序回归是一维保序回归的推广.本文给出了k=2,p=2时多维保序回归的求解方法. 令Xij,j=1,2,…,n是来自总体为二维正态分布N(ui,A)的样本,这里ui是未知的,A是已知的,i=1,2. 令μ=(μ1,μ2),Θ={(μ1,μ2)|μ1,μ2∈R2,)Θ0={(μ1,μ2)|μ1≤μ2,μ1,μ2∈R2},其中μ1≤μ2表示μ2-μ1的每一个分量为非负.本文也讨论了假设检验问题H0:μ∈Θ0,H1:μ∈Θ0=Θ-Θ0(H0是零假设).

Statistical inference under order restrictions is an important area of statistical analysis, isotonic regression theory plays a key role in this field. Multivariate isotonic regression are extersions of isotonic regression. This paper gives an algorithm of computiong the multivari-ate isotonic regression k = 2,p - 2.Let Xij,j =1,2, ..., n be independent 2-variate normal random vectors with unknown mean vectors ui and knowm covariance matrices A, i = 1,2. Let where μi ≤ μ2 means all the elements of μ2 -μ1 are nonnegative.The testing problem of hypotheses H0 :θ∈Θθ, H1:θ∈0=Θ-Θ0is also considered in this paper.