Counting and locating the solutions of polynomial systems of maximum likelihood equations, II: The Behrens-fisher problem

Max Louis G. Buot, Serkan Hoşten, Donald St P. Richards

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1343-1354
Number of pages12
JournalStatistica Sinica
Volume17
Issue number4
StatePublished - Oct 2007

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this