Toward Computerized Efficient Estimation in Infinite-Dimensional Models

ABSTRACT

Despite the risk of misspecification they are tied to, parametric models continue to be used in statistical practice because they are simple and convenient to use. In particular, efficient estimation procedures in parametric models are easy to describe and implement. Unfortunately, the same cannot be said of semiparametric and nonparametric models. While the latter often reflect the level of available scientific knowledge more appropriately, performing efficient inference in these models is generally challenging. The efficient influence function is a key analytic object from which the construction of asymptotically efficient estimators can potentially be streamlined. However, the theoretical derivation of the efficient influence function requires specialized knowledge and is often a difficult task, even for experts. In this article, we present a novel representation of the efficient influence function and describe a numerical procedure for approximating its evaluation. The approach generalizes the nonparametric procedures of Frangakis et al. and Luedtke, Carone, and van der Laan to arbitrary models. We present theoretical results to support our proposal and illustrate the method in the context of several semiparametric problems. The proposed approach is an important step toward automating efficient estimation in general statistical models, thereby rendering more accessible the use of realistic models in statistical analyses. Supplementary materials for this article are available online.

KEYWORDS:

• Asymptotic efficiency
• Canonical gradient
• Efficient influence function
• Nonparametric and semiparametric models
• Pathwise differentiability

Supplementary Material

In the Supplementary Material, we verify that the KL divergence and Hellinger distance satisfy conditions (B1), (B2) and (B3), and thus that they are appropriate divergences for the proposed representation of the EIF. We also verify directly that the proposed representation is valid in each of the four examples discussed in Section 5. Finally, we show that the use of an inappropriate divergence can lead to violations of the proposed representation. We do so by exhibiting a particular example based on parametric models and use of the L2 norm of the difference of density functions as divergence.

Additional information

Funding

The authors gratefully acknowledge the support of the Career Development Fund of the Department of Biostatistics at the University of Washington (MC), the New Development Fund of the Fred Hutchinson Cancer Research Center (ARL), and of NIH/NIAID grants 5UM1AI068635 (MC,ARL) and 5R01AI074345 (MJvdL).