On the Limiting Distribution of Roots of a Determinantal Equation
On the Limiting Distribution of Roots of a Determinantal Equation
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DOI：
10.1112/jlms/s1-16.3.183
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发表时间：
1941-07
影响因子：
1.2
通讯作者：
P. Hsu
中科院分区：
文献类型：
--
作者：
P. Hsu
I1= min（p，k-l），I2 = max（p，k-l），我们可以容易地看到，||ATJ||是正的并且秩为llt*，并且，如果N-k^ p，则矩阵||- w||是正定的 *。因此k中的行列式方程< f>l= 0（4）有一个重数为p-lx的零根和lx个真实的正根。方程（4）的非零的lx根在判别分析中起着重要的作用。它们的分布仅取决于（3）的根（见§ 3，第一段），在（3）的所有根为零的情况下，它们的精确分布是已知的。在本文中，我们得到的极限分布的最一般的情况下，作为样本大小成为无限的，而它们的比例保持不变。
I1= min (p, k—l), 12= max {p, k—1), we may readily see that the matrix|| atJ|| is positive and of rank llt* and that, provided that N—k^ p, the matrix||£ w|| is positive definite*. Hence the determinantal equation in< f> k,-<^ l= 0(4) has a root zero of multiplicity p—lx and lx real positive roots. The lx roots of (4) which are not zero play an important part in discriminant analysis f. Their distribution depends solely on the roots of (3)(see § 3, first paragraph) and their exact distribution is known% in the case where all the roots of (3) vanish. In this paper we obtain the limiting distribution for the most general case, as the sample sizes become infinite while their ratios remain constant.