# Estimation of unknown vector's amplitude with Gaussian noise

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)?

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 $$\mathbf{y}$$.