最大似然方程方程多项式组解的计数与定位，II：Bejers-Fisher问题
Counting and Locating the Solutions of Polynomial Systems of Maximum Likelihood Equations, II: The Behrens-Fisher Problem
摘要：
设$μ$是$p$维向量，$Σ_1$和$Σ_2$是$p \times p$正定协方差矩阵.在分别从独立的多变量正态总体$N_p(μ,Σ_1)$和$N_p(μ,Σ_2)$中获得大小为$N_1$和$N_2$的随机样本时，Behrs-Fisher问题是求解估计未知参数$μ$、$Σ_1$和$Σ_2$的似然方程。我们将证明，对于$N_1, N_2 > p$，几乎肯定存在似然方程的精确$2p+1$复解.对于$p = 2$的情况下，我们利用Monte Carlo模拟估计的相对频率与一个典型的Behrs-Fisher问题有多个真正的解决方案，我们发现，多个真正的解决方案很少发生。
英文摘要：
Let $μ$ be a $p$-dimensional vector, and let $Σ_1$ and $Σ_2$ be $p \times p$ positive definite covariance matrices. On being given random samples of sizes $N_1$ and $N_2$ from independent multivariate normal populations $N_p(μ,Σ_1)$ and $N_p(μ,Σ_2)$, respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters $μ$, $Σ_1$, and $Σ_2$. We shall prove that for $N_1, N_2 > p$ there are, almost surely, exactly $2p+1$ complex solutions of the likelihood equations. For the case in which $p = 2$, we utilize Monte Carlo simulation to estimate the relative frequency with which a typical Behrens-Fisher problem has multiple real solutions; we find that multiple real solutions occur infrequently.
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