Discussion of the Paper "Connecting Model-Based and Model-Free Approaches to Linear Least Squares Regression" by Lutz Dümbgen and Laurie Davies (2024): The Monotonicity Problem

Dmbgen and Davies (2024) provided analytical p-values for testing specific null hypotheses under linear models, which are of practical importance.In this discussion, we examine a feature of p-values that could lead to conflicting conclusions within the framework of linear models.The authors introduced the full model in their Equation (1) and demonstrated that the p-value to test the hypothesisis p-value(H p o , y) = B (n-p)/2,( p-p o )/2 y- 2 y-o 2 , where p o , B a,b , , and o are defined in the main paper.Consider two nested null hypotheses, namely, H p o 1 and H p o 2 , where