标题： 计算和定位多项式系统最大似然方程的解，II：Behrens-Fisher问题 显示英文标题

标题： Counting and Locating the Solutions of Polynomial Systems of Maximum Likelihood Equations, II: The Behrens-Fisher Problem

摘要： Let $\mu$ be a $p$-dimensional vector, and let $\Sigma_1$ and $\Sigma_2$ be $p \times p$ positive definite covariance matrices. 在分别从两个独立的多元正态总体 $N_p(\mu,\Sigma_1)$ 和 $N_p(\mu,\Sigma_2)$ 中抽取大小为 $N_1$ 和 $N_2$ 的随机样本时，Behrens-Fisher 问题旨在求解似然方程以估计未知参数 $\mu$、$\Sigma_1$ 和 $\Sigma_2$。 我们将证明，对于$N_1, N_2 > p$，几乎必然地恰好有$2p+1$个复数似然方程解。 当$p = 2$时，我们利用蒙特卡洛模拟来估计典型 Behrens-Fisher 问题具有多重实数解的相对频率；我们发现多重实数解发生的频率很低。 显示英文摘要

摘要： Let $\mu$ be a $p$-dimensional vector, and let $\Sigma_1$ and $\Sigma_2$ 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 $N_p(\mu,\Sigma_1)$ and $N_p(\mu,\Sigma_2)$, respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters $\mu$, $\Sigma_1$, and $\Sigma_2$. 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.