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

Max Louis G. Buot, Serkan Hoşten, Donald Richards

Research output: Contribution to journalArticle

14 Citations (Scopus)

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 1 2007

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Behrens-Fisher Problem
Polynomial Systems
Maximum Likelihood
Counting
Likelihood
Normal Population
Multivariate Normal
Positive definite
Unknown Parameters
Covariance matrix
Monte Carlo Simulation
Estimate
Polynomials
Maximum likelihood
Monte Carlo simulation

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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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.

In: Statistica Sinica, Vol. 17, No. 4, 01.10.2007, p. 1343-1354.

Research output: Contribution to journalArticle

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