关于二维正态总体的一个假设检验问题

ON A PROBLEM OF TESTING A CERTAIN STATISTICALHYPOTHESIS CONCERNING BIVARIATENORMAL POPULATIONS

§1.问题的提出 一种药品对于k种病菌有杀伤力,问对于哪一种病菌的效果最好? 设以在白鼠身上培养的病菌为例:给定N只白鼠,在每只鼠身上各取单位体积的血液,观察其中所含的K种病菌的数目,分别设为x11…x1N;…;xk1…xkN,（这里第一个足标是指哪一种病菌。

The following statistical hypothesis problem originates from medical studies:Samples(i=1,2,…,k)are supposed to come from populations Nand the problem, is to test thenull hypothesis H0, that the ratiosηi/ξi are the same for i = 1,2,…kFive cases are discussed in this paper.Case 1.By means of the likelihood ratio method, we take the rejection region θ>A here θ is the smallest root of the equationand attempt to choose A in order that the P{θ>A|H0}does not surpass a prescribed level a. It is found that this can be achieved when the constant A is determined by P{Xk2>A} = α.Case 2.In this case we may even suppose that Σ is dependent upon "i" i. e.the result is similar to case 1 except that θ is the smallest rootof the equationCaw 3,where σ is not known, We also make use of the likelihood ratio method and get the rejection region.in which θ is defined as ease 1, but the constant A is determined byHere again, we have P{θ>A|H0}≤α.Case 4. Σ is not a diagonal but is known.This case can easily be reduced to case 1. We get the rejection region θ1>A, where A is defined by a as in case 1, but θ is the smallest root of equationCase 5. Σ is not diagonal and is not known.Guided by the steps in case 4, we propose a rejection region {θ>A} where θ is the smallest root of the equationUsing a series of orthogonal transformations, we find two upperbounde for P{θ>A|H0}, viz.: where the θ1. is the largest root of the equation |VV'-θYY| =0 with(V,Y)-N((0),I). Therefore, if B and B' are defined by P{F(k,m)>B}-=α, P{θ1>B}=α, thenthe constant A=min{k/MB,B'} will satisfy the condition P{θ>A| H0}≤α.