In orthogonal regression it is assumed that both variables have noise. I'm interested in the simplest possible case. That is, I have a very large number of data points $(X_1,Y_1), ..., (X_n,Y_n)$. I know that $Y = a X + b$ and that my $X_i$ and $Y_i$ are observations of an underlying $(X,Y)$ pair, with uncorrelated Gaussian noise.
The standard approaches for processing this data, to estimate $a$ and $b$ assume the standard deviation of the noise in $X$ and $Y$ is identical. (Mathworks has one such presentation.)
My question: is there a way to estimate $\sigma_X$ and $\sigma_Y$? My current approach examines the noise in $Y$ and $X$ after the orthogonal fit. And see if they are commensurate. Seems like a reasonable approach, but if there's an approach with a better foundation, I'd be interested in hearing about it.