A Profile Wald Test In M-Estimation

Despite the growing popularity of machine learning-based inference, classical statistical inference remains highly relevant in modern data science due to its interpretability and theoretical rigor. Among its core tools, the likelihood ratio test, Wald test, and score test are foundational methods for hypothesis testing within the maximum likelihood framework. Although these tests are asymptotically equivalent under regularity conditions, each offers distinct advantages depending on the context, computational demands, and the availability of parameter estimates. In this dissertation, we introduce a fourth method, the Profile Wald Test (PWT), within the broader M-estimation framework. The PWT is based on profile estimators of the parameters under test, and occupies an intermediate position between the score test and the Wald test in terms of estimation requirements and computational efficiency. We establish the asymptotic properties of the PWT and assess its performance through extensive numerical studies. Our results show that the PWT not only maintains nominal type I error rates but also achieves statistical power comparable to the Ordinary Wald Test (OWT) when a convexity condition is satisfied. Additionally, its strong empirical performance and robustness make it a practical alternative to classical methods with broad applications in modern statistical inference.