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QT est is able to evaluate the maximum-likelihood based goodness-of-fit of any such convex polytope, within numerical accuracy, provided that 1) the polytope is full-dimensional in that it has the same dimension as the full probability space (in Figure 9, the 3D pyramid is full-dimensional in the 3D cube; see the Online Supplement for nonfull-dimensional examples), and provided that 2) the user gives the program a complete mathematical characterization of the polytope's mathematical structure.

In practice, this means that the researcher who wants to test a random preference model will first have to determine the geometric description of the model. If the polytope is full-dimensional, then they can test the model using QT est up to computational accuracy. In the Online Supplement, we provide the corresponding complete system of 784 nonredundant constraints for Random. In Figure 9, the shaded region is an irregular pyramid characterized by the constraints 0.