ESTIMATING EQUATIONS WITH NUISANCE PARAMETERS: THEORY AND APPLICATIONS

KE-HAI YUAN^{1} AND ROBERT I. JENNRICH^{2}

1Department of Psychology, University of California, Los Angeles, 1282A Franz Hall,

Box 951563, Los Angeles, CA 90095-1563, U.S.A.

2Department of Mathematics, University of California, Los Angeles, CA 90095, U.S.A.

(Received January 28, 1998; revised November 17, 1998)

Abstract. In a variety of statistical problems the estimate θ̂_{n} of a parameter θ is defined as the root of a generalized estimating equation G_{n}(θ̂_{n}, γ̂_{n}) = 0 where γ̂_{n} is an estimate of a nuisance parameter γ. We give sufficient conditions for the asymptotic normality of θ̂_{n} defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of θ̂_{n} will not be influenced by that of γ̂_{n} is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly.

Key words and phrases: Asymptotic distribution, generalized estimating equation, covariance structure analysis, pseudo maximum likelihood, generalized least squares, equivariant M-estimation.

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