The Annals of Statistics

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

Ying Ding and Bin Nan

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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.

Article information

Ann. Statist., Volume 39, Number 6 (2011), 3032-3061.

First available in Project Euclid: 24 January 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62N01: Censored data models
Secondary: 62D05: Sampling theory, sample surveys

Accelerated failure time model B-spline bundled parameters efficient score function semiparametric efficiency sieve maximum likelihood estimation


Ding, Ying; Nan, Bin. A sieve M-theorem for bundled parameters in semiparametric models, with application to the efficient estimation in a linear model for censored data. Ann. Statist. 39 (2011), no. 6, 3032--3061. doi:10.1214/11-AOS934.

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