On the MMSE Estimation of Norm of a Gaussian Vector under Additive White Gaussian Noise with Randomly Missing Input Entries

This paper considers the task of estimating the $l2$ norm of a $n$-dimensional random Gaussian vector from noisy measurements taken after many of the entries of the vector are \emph{missed} and only $K\ (0\le K\le n)$ entries are retained and others are set to $0$. Specifically, we evaluate the minimum mean square error (MMSE) estimator of the $l2$ norm of the unknown Gaussian vector performing measurements under additive white Gaussian noise (AWGN) on the vector after the data missing and derive expressions for the corresponding mean square error (MSE). We find that the corresponding MSE normalized by $n$ tends to $0$ as $n\to \infty$ when $K/n$ is kept constant. Furthermore, expressions for the MSE is derived when the variance of the AWGN noise tends to either $0$ or $\infty$. These results generalize the results of Dytso et al.\cite{dytso2019estimating} where the case $K=n$ is considered, i.e. the MMSE estimator of norm of random Gaussian vector is derived from measurements under AWGN noise without considering the data missing phenomenon.