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the maximum-likelihood estimate, satisfying an order-constrained model may lie on a face, an edge, or even a vertex of the shaded modal choice cube. This becomes even more complicated in higher dimensional spaces, where the modal choice model has surfaces of many different dimensions.
Rather, the distribution depends on the geometry of the model in question. The best fitting model parameters also need not be the orthogonal projection of the data onto the model in the geometric space. In sum, statistical testing of these models is difficult.