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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 approachthat also provide estimates for the errors of the estimated parameters?

thank you very much.

cheers

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# MLE is biased: are there other estimation methods?

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