The variables of interest are related by the following multivariate normal distribution: $$\begin{bmatrix} x \\ z_1 \\ z_2\end{bmatrix} \sim \mathcal{N}(\begin{bmatrix} \mu \\ \mu \\ \mu\end{bmatrix}, \begin{bmatrix} \sigma_{0}^2 & 0 & 0 \\ 0 & \sigma^2 & \rho \sigma^2\\ 0 & \rho \sigma^2 & \sigma^2\end{bmatrix})$$ $x$ is parameter, $z_1$ and $z_2$ are two noisy measurements. The minimum mean square error (MMSE) estimator of $x$ is $$\hat{x} = \mu + P_{xz}P_{zz}^{-1}(z - E[z])$$ where $z$ is measurement vector.

I want to know what $\sigma$ should be for estimator to be within $10\%$ of true value $x$ with $0.99$ probability.

Expressing problem statement mathematically I need to solve $\sigma$ from $$P\{|\frac{\hat{x} - x}{x}| < 0.1\} = 0.99$$

I can get approximate solution by replacing $x$ by $E[x]$ and using tables of cumulative normal distribution to get numerical value.

How good it that approximation?

Is it possible to get exact answer(numerical value, not closed from solution)?

Is it possible to calculate distribution for $\frac{\hat{x} - x}{x}$? At leas numerator and denominator are both Gaussian and dependent.


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