Statistics seminar

Abstract: Likelihood ratio tests (LRTs) are a central tool in statistics and data science for hypothesis testing, typically justified by Wilks’ theorem, which guarantees an asymptotic chi-square distribution under regular conditions. However, these conditions can fail in practice, and they do so in a surprisingly common class of problems: when parameters lie on the boundary of the parameter space. In these settings, the classical chi-square approximation is still often used, but it can be severely misleading, leading to incorrect inference and miscalibrated p-values. This talk introduces the geometric and probabilistic intuition behind this breakdown and explains how the asymptotic distribution of the LRT changes fundamentally.

I will present a concise derivation of the asymptotic distribution of the LRT under boundary conditions in the case of two parameters, and interpret it through the lens of tangent cones and projections of Gaussian random variables. I will then briefly discuss extensions that generalize these results to models with an arbitrary number of parameters on the boundary, including nuisance parameters.

The goal of the talk is to provide both theoretical insight and practical intuition for interpreting LRTs in constrained settings.