weight matrix $\mathbf{V}$, rank-deficiency of the information matrix is equivalent to rank-deficiency of the Jacobian of the sufficient statistics, $\dot{\boldsymbol{\sigma}}(\boldsymbol{\theta}) = \boldsymbol{\Delta}$. Because of this equivalence, and because the Jacobian has a much simpler form than the information matrix, we will focus here on the Jacobian.

When is the Jacobian rank deficient?
------------------------------------

We have seen that the Jacobian $\boldsymbol{\Delta}$ plays a central role in generating nonconvergence and “inadmissible” estimates (or bias), specifically when this matrix is rank-deficient. But when do such deficiencies occur? Here, we will distinguish two cases: underidentification and singular points.

Underidentification is the most well-known cause of rank deficiency of $\boldsymbol{\Delta}$. For example, when the degrees of freedom are negative, $p^* < p$ it is obvious that the $p^* \times p$ matrix $\boldsymbol{\Delta}$ will not have full column rank. The same occurs whenever the implied (co)variances of the model, and therefore the likelihood for which these are sufficient statistics, are equal for two different sets of parameter values, i.e. $\boldsymbol{\sigma}(\boldsymbol{\theta}) = \boldsymbol{\sigma}(\boldsymbol{\theta}^\prime)$ but $\boldsymbol{\theta} \neq \boldsymbol{\theta}^\prime$ [e.g. @wald1950note; @bekker_identification_2001]. The absence of this problem is referred to as “local” identification in the literature when the condition needs to hold only for an open neighborhood of the 