Likelihood-Neighborhood Calibration: Finite Likelihood-Ratio Secants for Parametric Estimation

Abstract

A finite-neighborhood estimating framework for regular parametric models is developed from exact change-of-measure identities between neighboring members of the same statistical family. Let f(x; θ) denote a regular one-parameter density and define the within-family likelihood ratio ( ) ( ) ( )
Whenever the two densities have common support, [ ( )]
holds exactly for every admissible finite perturbation h. The paper uses these identities systematically as unbiased estimating equations without requiring analytic differentiation of the likelihood. The resulting methodology is called Likelihood-Neighborhood Calibration Estimation (LNCE).
The paper develops both one-sided and symmetric likelihood-neighborhood estimators. The ordinary likelihood score is shown to arise as the zero-neighborhood tangent limit of the proposed likelihood-ratio secants, while the second and higher derivatives of the normalization identity recover the classical Bartlett identities. For a finite one-sided neighborhood, the Godambe information is expressed explicitly in terms of a Pearson divergence and a cross-score expectation. The exact relative efficiency is the squared correlation between the likelihood-ratio secant and the likelihood score and is therefore bounded above by one.
To eliminate dependence on the numerical scale of the parameter, an intrinsic version is introduced in which the neighboring models are selected at equal Kullback-Leibler distance from the candidate model. The resulting pair of neighboring probability laws is invariant under smooth one-to-one reparameterization. A symmetric KL-neighborhood likelihood secant converges in to the standardized score with second-order approximation error. Consequently, its asymptotic efficiency loss is of fourth order in the statistical neighborhood radius, whereas a one-sided neighborhood generally has a second-order efficiency loss.
Multiple neighborhood radii yield an overidentified estimating system. The optimally weighted estimator has Godambe information equal to the squared ( ) norm of the orthogonal projection of the likelihood score onto the span of the likelihood-neighborhood secants. Information increases monotonically as additional neighborhoods are added and converges to Fisher information when the system contains neighborhoods shrinking to zero.
A bounded likelihood-neighborhood construction is proposed for robust estimation. It preserves exact unbiasedness after model centering and retains reparameterization invariance under intrinsic neighborhoods. The associated influence function is bounded whenever the transformed likelihood-ratio contrast is bounded.
Two models admit particularly transparent theory. For normal location with known variance, the symmetric KL-neighborhood estimator has an explicit closed form and relative efficiency ( ) where r is the KL-standardized neighborhood radius. For the exponential rate model, a symmetric relative-neighborhood estimator is shown to have a unique root and fourth-order local efficiency loss. A reproducible Monte Carlo study confirms the local efficiency calculations. Under Gaussian contamination, raw finite likelihood-ratio secants are unstable, whereas the bounded KL-neighborhood estimator remains stable and competitive with standard Huber and density-power-divergence benchmarks.
The results provide a finite-neighborhood estimating-equation counterpart to classical score-based inference and organize finite within-family likelihood-ratio calibration within estimating-function, GMM, and robust-inference theory.