# 2-D parameter vector: Cramer Rao lower bound

Given a 2-D parameter vector, $\mathbf{X} = [x_1, x_2]$, let the corresponding $2\times2$ Fisher Information Matrix be $\mathbf{F}$. The Cramer-Rao Lower Bound (CRLB) is the inverse of the FIM. I obtain the following matrix by inverting $\mathbf{F}$ as: $$\mathbf{C} = \begin{bmatrix} \mathbf{C}_{1,1} & \mathbf{C}_{1,2}\\\mathbf{C}_{2,1} & \mathbf{C}_{2,2}\end{bmatrix}$$

I wanted to use a single error metric for both $x_1$ and $x_2$. I came across the following expression in one of the papers: $$f(\mathbf{X}) = \frac{1}{\sqrt{2}}\left(\mathbf{C}_{1,1}^2 + \mathbf{C}_{2,2}^2\right)^{1/2}$$

What does this quantity imply? Is this a joint CRLB for both parameters?

• I understand that $f(\mathbf{X})$ is actually the rms value of the lower bounds of both parameters. But I have not seen usage of this quantity before. – r2d2 Feb 1 '18 at 3:13
• Hi: If it was $\frac{1}{\sqrt n}$ rather $\frac{1}{\sqrt2}$, where $n$ was the sample size, then it would be the CRLB for the standard deviation of the sum of the 2 parameters. But, given that it's $\frac{1}{ \sqrt n}$, I can't help. maybe it's a typo ? – mark leeds Feb 1 '18 at 7:07