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i would appreciate any help you can offer for the following problem.

consider coordinates [itex]x_1, x_2[/itex] in the plane for which [itex]||x_1-x_2||=d[/itex].

suppose that this pair of coordinates can be measured independently, and that the measurements are 2D normally distributed with means [itex]x_1, x_2[/itex] and variances [itex]\sigma^2_1, \sigma^2_2[/itex]. given the value of [itex]d[/itex] and with known variances, how do i estimate the real position [itex]x_1, x_2[/itex] from a pair of measured positions?

the parameters to be estimated can be reduced to the coordinates of the center between [itex]x_1, x_2[/itex], giving 2 parameters (x,y coordinates), and an angle parameter to describe the angle of the vector [itex]x_1 - x_2[/itex]

with MLE, i have written down the probability density function, which takes the form of a 4-variate normal distribution with 3 unknown parameters. the extremal point of the log-likelihood in terms of [itex][/itex] can be written down explicitly.

the problem is that the solution seems to be biased, if the sqrt(variances) are on the order of [itex]d[/itex]. for very small variances it seems to work just fine.

do you know why this could be the case? and are there any alternatives to the MLE approach

**that also provide estimates for the errors of the estimated parameters**?

thank you very much.

cheers