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I have the following model: y = P v + n
Where y is the vector of observations, v is a unit vector and n is a Gaussian random noise whose covariance matrix is the identity matrix. P is a positive scalar. All the vectors are reals of size N.

I'd like to estimate the amplitude $P^2$ in the mse sense. All I know about v is that it's normalized and that each of its values has an equal chance of being positive or negative.

I can use the maximum likelihood for both P and v, getting the LS solution(in this case it's just setting the estimator to y), and then squaring it to get an estimate for $P^2$. A better estimate would be $y^T y -N$ which is unbiased and has a lower mse. Trading some variance for bias, we can improve that by setting the estimator to zero if $y^T y$ is smaller than N.

Is there a way to do better than this? If not, how can I prove this is the optimal solution ?
Can the ML be used in this case without directly estimating v (which is not needed)?

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1 Answer 1

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This is an old question but I'll throw together an answer anyway.

In statistical signal processing, its common to estimate $P^2$ via the leading eigenvalue of the covariance of $\mathbf{y}$. The argument goes as follows.

We assume that $P$ and $\mathbf{n}$ are mean zero, that $\mathbf{v}$ is fixed, and that $P$ and $\mathbf{n}$ are uncorrelated. Then some algebra shows that the covariance of $\textbf{y}$ is $$E[\mathbf{y} \mathbf{y}^{T}] = E[P^2] \mathbf{v} \mathbf{v}^{T} + \mathbf{I}.$$

The $N$ eigenvalues of this matrix in descending order are $E[P^2] + 1, 1, ..., 1$. The leading eigenvector is $\pm \mathbf{v}$ and the remaining eigenvectors are an orthogonal basis of the subspace in $\mathbb{R}^{N}$ perpendicular to $\mathbf{v}$. Therefore to get $E[P^2]$ we take the leading eigenvalue of $E[\mathbf{y} \mathbf{y}^{T}]$ and subtract 1.

Finally, we use a plug-in estimator for the parameters above by estimating $E[P^2]$ as one less than the leading eigenvalue of the sample covariance matrix of $\mathbf{y}$.

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