Asymptotic Efficiency and Le Cam Theory

Le Cam's theory makes precise what it means for an estimator to be asymptotically best, by approximating a smooth model near the truth with a simple normal experiment.

Definition

A regular estimator is asymptotically efficient if its limiting variance attains the lower bound set by the convolution and local-asymptotic-minimax theorems, equivalently the inverse Fisher information in a smooth parametric model.

Scope

This topic covers contiguity and Le Cam's lemmas, the local asymptotic normality of smooth parametric models, the limiting Gaussian shift experiment, the convolution theorem of Hajek showing that any regular estimator's limit is the efficient one plus independent noise, the local-asymptotic-minimax theorem, the consequent definition of asymptotic efficiency, and the role of the efficient influence function and superefficiency.

Core questions

• What is local asymptotic normality, and why does it reduce a model to a normal experiment?
• How does the convolution theorem characterize the best possible limiting distribution of an estimator?
• What does the local-asymptotic-minimax theorem add about worst-case risk?
• Why is superefficiency possible only on a negligible set, and what is the efficient influence function?

Key theories

Local asymptotic normality

For smooth models the log-likelihood ratio along local parameter perturbations behaves like that of a Gaussian shift experiment, so questions about the original model reduce to a tractable normal problem.

Convolution and local-asymptotic-minimax theorems

Hajek's convolution theorem shows the limit law of any regular estimator is the efficient normal law convolved with independent noise, and the local-asymptotic-minimax theorem bounds worst-case local risk, jointly defining asymptotic efficiency.

Clinical relevance

Le Cam's theory provides the benchmark of asymptotic efficiency against which estimators are judged and underlies the construction of efficient and semiparametric-efficient estimators, including the influence-function methods used in causal inference and targeted learning.

History

Le Cam developed contiguity and local asymptotic normality from the 1950s, resolving longstanding puzzles such as superefficiency. Hajek proved the convolution and local-asymptotic-minimax theorems around 1970, and the framework was extended to semiparametric models later in the century.

Key figures

• Lucien Le Cam
• Jaroslav Hajek
• Aad van der Vaart
• Peter J. Bickel

Seminal works

• vanderVaart1998

Frequently asked questions

What is superefficiency?

It is the phenomenon, illustrated by Hodges's example, of an estimator beating the efficient asymptotic variance at isolated parameter values; the convolution theorem shows this can happen only on a set of measure zero and at the cost of worse behavior nearby.

Why approximate a model by a normal experiment?

Because the limiting Gaussian shift experiment is fully understood, so optimality questions that are intractable in the original model can be answered there and transferred back via local asymptotic normality.