I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My supervisor insists I'm wrong (I'm sure I must be), but I can't seem to figure out what I'm doing wrong. Can someone please help guide me in the right direction? Sorry for the long development of the problem below.
A transmitter T is at an unknown location $\theta=(x, y)^{T}$. There are $s$ sensors in the region $\Omega$. The $j^{th}$ sensor positioned at $\theta_{j}=(x_{j}, y_{j})^{T}$ measures a distance/range to T that is a random variable dependent on the actual distance of T from the sensor ($d_{j}$).
The measurement vector for all anchors is therefore: \begin{align} r=[d_{1}+\epsilon_{1},...,d_{j}+\epsilon_{j}]^{T}=d(\theta)+\epsilon=\|\theta-\theta_{j}\|_{_{2}} + \epsilon \end{align}
where the error is assumed to be normally distributed $\epsilon \sim N\left(0, \Sigma_{s_{\mathrm{x}}s}\right)$.
For a multivariate Gaussian distribution, the $(m,n)$ element of the FIM is:
\begin{align} FIM_{m,n}=\frac{\partial d^{T}}{\partial \theta_m} \Sigma^{-1} \frac{\partial d}{\partial \theta_n}+\frac{1}{2} TR\left(\Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_m} \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_n}\right) \end{align}
Assuming independent noise at the sensor measurement and a constant variance for the noise at each sensor, the covariance matrix is diagonal, i.e. $\Sigma=diag(\sigma^{2}_{1}, \sigma^{2}_{2},..., \sigma^{2}_{s})$. Thus, the FIM can be expanded for $\theta=(x, y)^{T}$ as:
\begin{equation} \left[\begin{array}{cc} FIM_{1,1} & FIM_{1,2} \\ FIM_{2,1} & FIM_{2,2} \end{array}\right] \end{equation}
\begin{align} \label{FIM11} FIM_{1,1} &= \sum_{j=1}^{s} \frac{1}{2}\left(\sigma_{j}^{2}\right)^{-2}\left(\frac{\partial \sigma_{j}^{2}}{\partial x}\right)^{2}+\left(\sigma_{j}^{2}\right)^{-1}\left(\frac{\partial d_{j}}{\partial x}\right)^{2}\\ &= \sum_{j=1}^{s} \left(\sigma_{j}^{2}\right)^{-1}cos^{2}(\phi_j) \end{align}
\begin{align} \label{FIM12} FIM_{1,2}=FIM_{2,1}&=\sum_{j=1}^{s} \frac{1}{2}\left(\sigma_{j}^{2}\right)^{-2} \frac{\partial \sigma_{j}^{2}}{\partial x} \frac{\partial \sigma_{j}^{2}}{\partial y}+\left(\sigma_{j}^{2}\right)^{-1} \frac{\partial d_{j}}{\partial x} \frac{\partial d_{j}}{\partial y}\\ &=\sum_{j=1}^{s} \left(\sigma_{j}^{2}\right)^{-1} cos(\phi_j) sin(\phi_j) \end{align}
\begin{align} \label{FIM22} FIM_{2,2} &= \sum_{j=1}^{s} \frac{1}{2}\left(\sigma_{j}^{2}\right)^{-2}\left(\frac{\partial \sigma_{j}^{2}}{\partial y}\right)^{2}+\left(\sigma_{j}^{2}\right)^{-1}\left(\frac{\partial d_{j}}{\partial y}\right)^{2}\\ &=\sum_{j=1}^{s} \left(\sigma_{j}^{2}\right)^{-1}sin(\phi_j)^{2} \end{align}
Where $\phi_j$ is the angle between the $j^{th}$ sensor and T.
Consider 3 sensors situated at (-1,0),(0,0) and (1,0), and T at (2,0). The FIM is clearly singular as $\phi_j=0 \forall j$ and so the CRB does not exist.
The circles for two different range measurements are shown, and overlayed with the negative log-likelihood (NLL) function for the measurement. When 2 or more circles intersect, they do so at 2 locations which have reflection symmetry along the y=0 line, and so the NLL function has a minima at these two points. When no circles intersect, the NLL function has a minima on the y=0 line. Therefore, the y-coordinate MLE (i.e. estimator that minimizes NLL function) is unbiased and has finite variance.
This is the output of 1500 MLE estimates I computed in a simulation:
I can't seem to figure out where I've gone wrong.