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How is the minimum variance unbiased estimator derived for some RV $X\sim \mathcal{N}(0,\sigma^2)$, given $\mathbf{Y}=X\cdot \mathrm{ones}_{N\times 1}+\mathbf{w}$, where $\mathbf{w}$ is $N$-dimensional multivariate-normal with mean $[0, \dots, 0]^T$ and covariance matrix $\mathbf{\Sigma}$, and is independent of $X$?

In words, what is the best estimator for a signal in correlated noise, what is the distribution?

I know the answer is well-known and buried somewhere in Kay's Estimation Theory text but I cannot find it at the present. (am still looking)


If $\mathbf{\Sigma}$ were diagonal (that is, uncorrelated noise), we know that this estimator is some linear combination of the $\mathbf{Y}$ and it's Gaussian distributed about the signal with variance $\sum_{n=1}^N \frac{1}{\mathbf{\Sigma}_{n,n}}$ (cf. Maximal-ratio combining)

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From Kay's Fundamentals of Statistical Signal Processing, Volume I:

$$\textbf{Example 4.4 - DC Level in Colored Noise}$$ $\text{We now extend Example 3.3 to the colored noise case. If }x[n]=A+w[n]\text{ for }n=0,1,\dots,N-1,\text{ where }w[n]\text{ is }\textit{colored}\text{ Gaussian noise with }N\times N\text{ covariance matrix }\mathbf{C},\text{ it immediately follows from (4.25) that with }\mathbf{H}=\mathbf{1}=[1\ 1\dots 1]^T, \text{the MVU estimator of the DC level is}$ \begin{array}\hat{A} &=& (\mathbf{H}^TC^{-1}\mathbf{H})^{-1}\mathbf{H}^TC^{-1}\mathbf{x} \\ &=&\displaystyle \frac{\mathbf{1}^T\mathbf{C}^{-1}\mathbf{x}}{\mathbf{1}^T\mathbf{C}^{-1}\mathbf{1}}\end{array} $\text{and the variance is, from (4.26),}$ $$\begin{array}\displaystyle \text{var}(\hat{A}) &=& (\mathbf{H}^TC^{-1}\mathbf{H})^{-1} \\ &=& \displaystyle \frac{1}{\mathbf{1}^T\mathbf{C}^{-1}\mathbf{1}}. \end{array}$$

iThe whole series is a great reference.

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