A sieve M-theorem for bundled parameters in semiparametric models, with application to the efficient estimation in a linear model for censored data

TL;DR

This paper develops a general sieve M-theorem for bundled parameters in semiparametric models, enabling efficient estimation in complex models like censored linear regression, with theoretical guarantees and practical advantages over existing methods.

Contribution

It introduces a novel sieve M-theorem for bundled parameters and applies it to derive efficient estimators in censored linear models, enhancing estimation accuracy.

Findings

Estimator is consistent and asymptotically normal.

Achieves the semiparametric efficiency bound.

Performs better than existing methods in simulations.

Abstract

In many semiparametric models that are parameterized by two types of parameters---a Euclidean parameter of interest and an infinite-dimensional nuisance parameter---the two parameters are bundled together, that is, the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument. For example, in a linear regression model for censored survival data, the unspecified error distribution function involves the regression coefficients. Motivated by developing an efficient estimating method for the regression parameters, we propose a general sieve M-theorem for bundled parameters and apply the theorem to deriving the asymptotic theory for the sieve maximum likelihood estimation in the linear regression model for censored survival data. The numerical implementation of the proposed estimating method can be achieved through the conventional gradient-based search algorithms such as the Newton--Raphson algorithm. We show that the proposed estimator is consistent and asymptotically normal and achieves the semiparametric efficiency bound. Simulation studies demonstrate that the proposed method performs well in practical settings and yields more efficient estimates than existing estimating equation based methods. Illustration with a real data example is also provided.