AISM 52, 343-350

## Estimating equations with nuisance parameters: Theory and applications

### Ke-Hai Yuan^{1} and Robert I. Jennrich^{2}

^{1}Department of Psychology, University of
California, Los Angeles, 1282A Franz Hall, Box 951563, Los Angeles,
CA 90095-1563, U.S.A.

^{2}Department 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 *G*_{n}(\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.

**Source**
( TeX , DVI )