Conventional tests for composite hypotheses in minimum distance models can be unreliable when the relationship between the structural and reduced‐form parameters is highly nonlinear. Such nonlinearity may arise for a variety of reasons, including weak identification. In this note, we begin by studying the problem of testing a “curved null” in a finite‐sample Gaussian model. Using the curvature of the model, we develop new finite‐sample bounds on the distribution of minimum‐distance statistics. These bounds allow us to construct tests for composite hypotheses which are uniformly asymptotically valid over a large class of data generating processes and structural models.