# Estimating (MLE) 2D Vector Entries by a Noisy Samples of its Entries and its Norm

I'd like your assistance with developing the Maximum Likelihood Estimator (MLE) and CRLB of the following case:

Given a 2D vector (A Point in XY Plane) $p = ({x}_{p}, {y}_{p})$.
The measurements are noisy measurements of this vector entries and its norm.
Namely the measurements vector is $m = ({x}_{m}, {y}_{m}, {r}_{m})$.
Where the the distribution of those are given by:

$${x}_{m} \sim N({x}_{p}, {\sigma}_{x}), {y}_{m} \sim N({y}_{p}, {\sigma}_{y}), {r}_{m} \sim N(\sqrt{{x}_{p}^{2} + {y}_{p}^{2}}, {\sigma}_{r})$$

Namely, the measurements are Normally Distributed, Unbiased and Independent of each other.

The parameters to estimate are $\Theta = ({x}_{p}, {y}_{p})$ given the measurement vector $m = ({x}_{m}, {y}_{m}, {r}_{m})$ as defined above.

In simple words, Estimate a point coordinates in the 2D plane given a noisy measurements of its coordinates and its range / distance from (0, 0).

I'd be happy to hear your solutions, ideas, related articles and the CRLB (Or any other lower bound on the estimation).

• Fascinating question. Is there only a single observation-triplet, or is this for a sample of size $n$, say? – Glen_b -Reinstate Monica Nov 5 '13 at 15:11
• @Glen_b, Let's start with the simple case, only 1 triplet of measurements. Thank You. – Royi Nov 5 '13 at 15:26
• Are the $\sigma$'s known? – Glen_b -Reinstate Monica Nov 6 '13 at 4:18
• Yep, the STD'd are given and known. And you know the distribution is Normal. You can see the unknown are the expected value (Part of). – Royi Nov 6 '13 at 5:25