AISM 52, 343-350

## Estimating equations with nuisance parameters: Theory and applications

### Ke-Hai Yuan1 and Robert I. Jennrich2

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 \hat{theta}n of a parameter theta is defined as the root of a generalized estimating equation Gn(\hat{theta}n, \hat{gamma}n) = 0 where \hat{gamma}n is an estimate of a nuisance parameter gamma. We give sufficient conditions for the asymptotic normality of \hat{theta}n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of \hat{theta}n will not be influenced by that of \hat{gamma}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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