Semiparametrically Efficient Estimation of Constrained Euclidean Parameters

TL;DR

This paper develops a method for efficient estimation of Euclidean parameters constrained to a smooth surface within semiparametric models, extending efficiency results to constrained settings and applying them to various models.

Contribution

It introduces a general approach for efficient estimation of constrained Euclidean parameters in semiparametric models, utilizing the chain rule for score functions and demonstrating broad applicability.

Findings

Efficient score functions for constrained parameters are derived using the chain rule.

The method achieves semiparametric efficiency for the constrained parameters.

Applications include location-scale, Gaussian copula, and regression models.

Abstract

Consider a quite arbitrary (semi)parametric model 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 original estimator, 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 under linear restrictions.