### Abstract

Let μ be a p-dimensional vector, and let Σ_{1} and Σ_{2} be p × 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 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.

Original language | English (US) |
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Pages (from-to) | 1343-1354 |

Number of pages | 12 |

Journal | Statistica Sinica |

Volume | 17 |

Issue number | 4 |

State | Published - Oct 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Statistica Sinica*,

*17*(4), 1343-1354.

}

*Statistica Sinica*, vol. 17, no. 4, pp. 1343-1354.

**Counting and locating the solutions of polynomial systems of maximum likelihood equations, II : The Behrens-fisher problem.** / Buot, Max Louis G.; Hoşten, Serkan; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Counting and locating the solutions of polynomial systems of maximum likelihood equations, II

T2 - The Behrens-fisher problem

AU - Buot, Max Louis G.

AU - Hoşten, Serkan

AU - Richards, Donald

PY - 2007/10/1

Y1 - 2007/10/1

N2 - Let μ be a p-dimensional vector, and let Σ1 and Σ2 be p × p positive definite covariance matrices. On being given random samples of sizes N1 and N2 from independent multivariate normal populations Np(μ, Σ1) and Np(μ, Σ2), respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters μ, Σ1, and Σ2. We prove that for N1, N2 > 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.

AB - Let μ be a p-dimensional vector, and let Σ1 and Σ2 be p × p positive definite covariance matrices. On being given random samples of sizes N1 and N2 from independent multivariate normal populations Np(μ, Σ1) and Np(μ, Σ2), respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters μ, Σ1, and Σ2. We prove that for N1, N2 > 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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M3 - Article

AN - SCOPUS:38549130370

VL - 17

SP - 1343

EP - 1354

JO - Statistica Sinica

JF - Statistica Sinica

SN - 1017-0405

IS - 4

ER -