identifiable even though $\text{rk}(\Delta)<p$. This is possible because both functions (the example function in Figure \[fig:fx\] and the fitting function for SEM) have a discontinuous second derivative (Hessian) at the optimum.

While the above may seem like happy news, underidentification is not the only problem that can be caused by rank deficiencies. This can be seen by generating data from the model given above with the rank deficiency and fitting the model with free parameters to these data. For illustration purposes, I used $\lambda_1 = 1$, $\lambda_1 = 0.4$, $\lambda_3 = 0.7$, $\psi_1 = 1$, $\psi_2 = 0.3$, and $\psi_3 = 0$ as parameter settings and generated 2000 datasets with 100,000 cases each.

The results are shown in Figure \[fig:shapiro\] using maximum-likelihood (top panel) and a single run of Hamiltonian Monte Carlo using `stan` (bottom panel). In both cases the results lead to estimation problems. Using ML, the estimates $\hat{\psi}_3$ do not concentrate symmetrically around the true value, $\psi_3 = 0$. In spite of the very large number of cases, due to the rank deficiency, there is a nonzero probability that $\psi_3$ takes on an arbitrarily large value. For those solutions that are near the true value (spike near zero), 