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Find estimator for bivariate distribution

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Luca Amerio
Luca Amerio on 2 Feb 2017
Hi I have a set of Joint Probability Density Functions obtain from some sets of data. I would like to find the "best fitting line" that describes the distribution. In other words, I would like the dashed line in the figure.
One important point is that the two variables are both affected by error, therefore the estimator should be "simmetric", that is return the same line if applied f(x,y) or f(y,x) (mirrored of course).
I tried several estimators, but couldn't find what I'm looking for. Both the Ordinary Least Square (polyfit) and the Generalised Least Square (mvregress) returns a results that depends by the order of the inputs. This is also very clear in the help of mvregress. I found this equation
from the wikipedia page Multivariate normal distribution. This is indeed simmetric in x and y, but it returns the dark-red line in the figure, not bad, but nor what I would expect. I think it is due to the face that the distribution is far from being normal.
Do you have any suggestion to find a good estimator for this problem?
Thank you very much

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