a simple classical example discussed by @shapiro_investigation_1983.

@shapiro_investigation_1983 stressed that rank-deficiency of the information matrix need not imply a non-identified model. Figure \[fig:fx\] demonstrated how this is possible: a function may well have a unique maximum even though it does not have a full-rank second derivative at every point. In this case, as noted by @shapiro_investigation_1983, the regularity conditions suggested by @wald1950note are violated, as the second derivative is not constant within any neighborhood around the maximum, but changes abruptly when moving away from this point (third panel in Figure \[fig:fx\]). For this reason, the rank condition on the information matrix is only indicative of a true identification problem when this regularity condition is met.

@shapiro_investigation_1983 illustrated this point with a three-indicator confirmatory factor model reparameterized as $\boldsymbol{y} = \Lambda \boldsymbol{\eta}$ with $$\begin{aligned}
    \Lambda = \kbordermatrix{
      &  \eta & \epsilon_1 & \epsilon_2 & \epsilon_3 \\
       y_1 &  \lambda_1 &  \psi_1 & 0 & 0\\
       y_2 &  \lambda_2 & 0 & \psi_2 & 0 \\
         y_3 &\lambda_3 & 0 & 0 & \psi_3 \\
    }, & &
    \Phi = \text{Var}(\boldsymbol{\eta}) = \mathbf{I}_4.\end{aligned}$$ All implied variances and covariances can then be written $\sigma_{jj^\prime}(\boldsymbol{\theta}) = \lambda_j \lambda_{j^\prime} + \delta_{j j^\prime} \psi_j^2$, where $\delta_{j 