# Alternatives to calculating the rank of the information matrix in determining if the model is identifiable

I have a known non-linear model $$h \in \mathbf{R}^n$$: $$y = h(\theta) + \epsilon,$$ where $$\theta\in \mathbf{R}^m$$ is a parameter vector, and $$\epsilon$$ is a normal random variable with zero mean and known covariance. Thus, the distribution of $$y$$ is $$\mathcal{N}(h(\theta), \Sigma$$). To determine if the model $$h$$ is identifiable, one can use,

Rothenberg, Thomas J. “Identification in Parametric Models.” Econometrica, vol. 39, no. 3, 1971, pp. 577–591. JSTOR, JSTOR, www.jstor.org/stable/1913267.

and Theorem 1 states that if the information matrix is non-singular in $$\theta_0$$, then the model is locally identifiable at $$\theta_0$$. The information matrix in this case can be calculated as $$I(\theta) = \left(\frac{\partial h}{\partial \theta} \right)^T \Sigma^{-1}\left(\frac{\partial h}{\partial \theta} \right),$$ and to know if $$I$$ is singular one can attack $$I$$ directly, or look at the rank of the Jacobian.

Now, if the function $$h$$ is complicated, the Jacobian may be even more complicated, and $$I$$ is hard to analyse. Are there any alternatives to analyse identifiability of a model other than looking at the singularity of $$I$$?