I unable to solve item 2 from the problem posted below, solution or tips are appreciated.
My attempt
I read item 2 as $$ P\{|\frac{\hat{x} - x}{x}| < 0.1\} = 0.99 \tag{1}\label{base} $$ where $\hat{x} = \bar{x} - P_{xz}P_{zz}^{-1}(z-\bar{z})$ is MMSE estimator. $z = \begin{bmatrix}z_1 \\ z_2\end{bmatrix}$ is random vector of measurements, $P_{xz}$ is cross-covariance of $x$ and $z$, $P_{zz}$ is covariance of $z$, and $\bar{z}$ is expectation of $z$.
$$ P_{xz} = \sigma_{0}^2 \begin{bmatrix}1 & 1\end{bmatrix} $$
$$ P_{zz} = E[(z - \bar{z})(z - \bar{z})'] = \begin{bmatrix} \sigma_{0}^2 + \sigma^2 & \sigma_{0}^2 + \rho \sigma^2 \\ \sigma_{0}^2 + \rho \sigma^2 & \sigma_{0}^2 + \sigma^2 \end{bmatrix} $$
$$ \bar{z} = \bar{x} $$
Evaluating $\sigma$ from $\eqref{base}$ is doubtful. For one thing $\frac{\hat{x} - x}{x}$ is not Gaussian. I though that it might be acceptable replace $x$ by $\bar{x}$ in denominator and calculate $$ P_{\hat{x}|z}\{|\frac{\hat{x} - x}{\bar{x}}| < 0.1\} = 0.99 $$
$$
P_{\hat{x}|z}\{|\frac{\hat{x} - x}{\sqrt{P_{xx|z}}}| < \frac{0.1\bar{x}}{\sqrt{P_{xx|z}}}\} = 0.99 \tag{2}\label{aprx}
$$
With assumption that $x$ is expected value of $\hat{x}$ conditioned on measurements.
$\eqref{aprx}$ is standardization of $\hat{x}$. Using Normal distribution probabilities table
$$ \frac{0.1\bar{x}}{\sqrt{P_{xx|z}}} = 2.58 \Leftrightarrow \Big(\frac{0.1\bar{x}}{2.58}\Big)^{2} = P_{xx|z} = \frac{(1+\rho) \sigma_{0}^{2} \sigma^{2}}{2\sigma_{0}^{2} + (1+\rho) \sigma^{2}} \tag{3}\label{final} $$
From $\eqref{final}$ I can calculate $\sigma=0.48$. To check my solution I simulated estimation for 10000 times and get $1\%$ accuracy. My solution is too low. By simulating with different values $\sigma=0.9$ is correct value.
Either my simulation has errors or my solutions has errors.
Problem description
Given the prior information $x \sim \mathcal{N}(\bar{x}, \sigma_{0}^2)$ and the measurements $$ z(j) = x + w(j) \quad j=1,2\ $$ with the jointly Gaussian measurements noises $w(j) \sim \mathcal{N}(0, \sigma_{0}^2)$ independent of $x$ but correlated among themselves, with $$ E[w(1)w(2)] = \rho \sigma^2 $$
- Find the variance of the MMSE estimator of $x$ conditioned on these measurements.
- If $\bar{x}=10$, $\sigma_0=1$ and $\rho=0.5$, how accurate should the measurements be (i.e., find $\sigma$) if we want the estimate to be within $10\%$ of the true value with $99\%$ probability.
Source:
Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software
Yaakov Bar-Shalom, X. Rong Li, Thiagalingam Kirubarajan