An invariant modification of the bilinear form test

2026-02-05

AI Summary

This study addresses the lack of invariance of bilinear form tests under parameter reparameterization, which leads to inconsistent hypothesis testing results in extremum estimation. For the first time, the paper establishes sufficient conditions under which such tests remain invariant to reparameterization and constructs a corrected test statistic accordingly. The proposed method integrates principles of invariance from M-estimation, estimating functions, and generalized method of moments theory, ensuring consistent inference across different parameterizations. Monte Carlo simulations demonstrate that the corrected statistic exhibits favorable small-sample properties and robustness across various settings, effectively resolving reliability issues arising from parameter transformations in practical applications.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The invariance properties of certain likelihood-based asymptotic tests as well as their extensions for M-estimation, estimating functions and the generalized method of moments have been well studied. The simulation study reported in Crudu and Osorio [Econ. Lett. 187: 108885, 2020] shows that the bilinear form test is not invariant to one-to-one transformations of the parameter space. This paper provides a set of suitable conditions to establish the invariance property under reparametrization of the bilinear form test for linear or nonlinear hypotheses that arise in extremum estimation which leads to a simple modification of the test statistic. Evidence from a Monte Carlo simulation experiment suggests good performance of the proposed methodology.

Problem

• invariance
• bilinear form test
• reparametrization
• extremum estimation
• hypothesis testing

Innovation

• bilinear form test
• invariance
• reparametrization
• extremum estimation
• M-estimation