Effect of imprecisely known nuisance parameters on estimates of primary parameters

Let the model under consideration have two types of parameters, α and θ, where α represents the nuisance parameters and θ represents the primary parameters (or parameters of interest). We will suppose that θ is estimated after α has been fixed at some value, and that is determined from information other than that used to form the estimate for θ. This paper examines the properties of in the presence of uncertainty in the assumed value for ∝. In particular, we consider: (1) the effect of the uncertainty in on the dispersion matrix, and hence confidence intervals, for ; and (2) the effect of on the asymptotic properties of . The results of this paper are developed in a multivariate setting and apply to a broad class of estimators, including maximum likelihood, maximum a posteriori, and least squares.