Asymptotic efficiency and convolution bounds

Let the target be a differentiable scalar function of a finite-dimensional parameter. An estimator is regular at the reference point if, under every fixed local displacement, its root-sample-size error centered at the displaced target converges to the same limiting law. Centering at the moving target is essential. An estimator's distribution under the reference law alone cannot establish this property. The definition makes performance stable across alternatives that remain statistically comparable at the LAN scale.

For an interior dominated iid model with differentiability in quadratic mean, nonsingular information, and the usual LAN structure, the finite-dimensional convolution theorem constrains any such regular estimator with a limiting distribution. Its limit is the convolution of the efficient centered Gaussian with another probability law. The two components can be realized independently. If the limiting law has finite second moment, its variance is at least the efficient Gaussian variance. Convergence of actual scaled finite-sample variances requires extra uniform integrability; weak convergence alone does not give that conclusion.

For a regular asymptotically linear estimator, an observation-level influence function has mean zero and finite second moment. Its covariance with the model score equals the derivative of the target. The efficient influence function is the target derivative times inverse information times the score. Subtracting it from any other valid influence function leaves a residual orthogonal to the score. Orthogonality splits the variance into efficient variance and a nonnegative remainder; the joint central limit theorem makes the corresponding Gaussian components independent.

This argument applies within its stated asymptotically linear class, whereas the convolution theorem also covers regular limits that are not introduced through an explicit influence expansion. Equality in the variance bound means no additional limiting variance, provided those variances exist. It does not by itself compare finite-sample bias, numerical cost, robustness to model misspecification, or higher-order coverage. Choosing an estimator still involves those questions, but an efficiency claim should identify the model and the target whose information bound is being attained.

In a normal location model, a Hodges-type estimator sets the sample mean to zero whenever its magnitude is at most the inverse fourth root of sample size, and otherwise uses the sample mean. At the zero parameter, its root-sample-size error converges to zero. Yet under a local mean equal to a fixed nonzero displacement divided by the square root of sample size, it still selects zero with probability tending to one, so the error centered at the local target tends to the negative displacement.

The limit therefore depends on the local displacement and violates regularity. This does not refute the convolution bound; it explains why that condition is present. A pointwise variance comparison at zero would hide the unstable neighboring behavior. When assessing a claimed improvement over information, inspect the target's differentiability, the local centering, the estimator's regularity, and the existence of the relevant moments. A single favorable simulation at one parameter cannot settle these structural requirements.