Consider a quite arbitrary (semi)parametric model for i.i.d. observations with a Euclidean parameter of interest and assume that an asymptotically (semi)parametrically efficient estimator of it is given. If the parameter of interest is known to lie on a general surface (image of a continuously differentiable vector valued function), we have a submodel in which this constrained Euclidean parameter may be rewritten in terms of a lower-dimensional Euclidean parameter of interest. An estimator of this underlying parameter is constructed based on the given estimator of the original Euclidean parameter, and it is shown to be (semi)parametrically efficient. It is proved that the efficient score function for the underlying parameter is determined by the efficient score function for the original parameter and the Jacobian of the function defining the general surface, via a chain rule for score functions. Efficient estimation of the constrained Euclidean parameter itself is considered as well.
Our general estimation method is applied to location-scale, Gaussian copula and semiparametric regression models, and to parametric models.
"Semiparametrically efficient estimation of constrained Euclidean parameters." Electron. J. Statist. 11 (2) 3120 - 3140, 2017. https://doi.org/10.1214/17-EJS1308