\[fig:fx\]. The Figure illustrates the function $f(x) = \text{sign}(x) \cdot x^3$. In this example, $f(x)$ plays the role of the (log)“likelihood”, $x$ is the parameter, the first and second derivatives (second and third panels) are the gradient and Hessian, and the fourth panel plots the inverse second derivative, which plays the role of the variance of the parameter estimate. It can be seen in Figure \[fig:fx\] that, while the second derivative is rank-deficient at the point $x=0$, which also happens to be the maximum, this maximum is still uniquely identifiable. However, as the point is approached, the variance of any finite-sample estimate of the parameter will grow without bound (asymptote in fourth panel).

![Example of a rank-deficient function, $f(x) = \text{sign}(x) \cdot x^3$. The function has a unique maximum at $x=0$, but at this point its second derivative is zero and changes discontinuously. The rightmost panel shows the negative inverse second derivative, which plays the role of the variance in likelihood theory, has an asymptote at this point.[]{data-label="fig:fx"}](fx-derivatives-4.pdf){width="\textwidth"}

Figure \[fig:fx\] shows that estimation problems do not only occur at the “almost surely” impossible point $x=0$. Points *close* to this “point of deficiency” (p.o.d.) will also generate extremely high variance of 