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I have a question about a model related to Deming regression and would appreciate some help and/or publications to further study this model.

Statistical Model: \begin{align} X_{i,j}&=\mu_{X,i}+\varepsilon_{i,j},\quad \varepsilon_{i,j}\sim\mathcal{N}(0,\sigma^2_X)\\ Y_{i,j}&=a+b\mu_{X,i}+\tau_{i,j},\quad \tau_{i,j}\sim\mathcal{N}(0,\sigma^2_Y)\\ \end{align} with $\tau_{i,j}$ and $\varepsilon_{i,j}$ independent and $i=1,\ldots,n$, $j=1,\ldots,n_{i}$. Moreover, $\mu_{X,i}$ is non-random and unknown.

For $n_1=\ldots=n_n=1$ this model corresponds to the well-known Deming regression which has a solution for a known and fixed variance ratio $\delta = \sigma^2_Y/\sigma^2_X$. In my case though, $n_i$ is at least two.

Questions:

  1. Could someone recommend good resources (publications preferred) about statistical inference in the model described above?
  2. I tried to calculate the maximum-likelihood estimators for $a$, $b$, $\sigma^2_X$, $\sigma^2_Y$, and $\mu_i$ through maximizing the log-likelihood function \begin{align} -\sum_i\sum_{j=1}^{n_{i}}\left(\frac{(X_{i,j}-\mu_{X,i})^2}{2\sigma^2_X}+\log(\sigma^2_X)+ \frac{(Y_{i,j}-b\mu_{X,i}-a)^2}{2\sigma^2_Y}+\log(\sigma^2_Y)\right) \end{align} However, I couldn't solve the resulting equations. Does anybody have similar experiences?
  3. Is it even possible to get a solution without knowing $\delta$?
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    $\begingroup$ +1 I don't have a proper answer, but I'm just starting to look at Deming regression and I'm finding the following paper by ET Jaynes to be highly illuminating bayes.wustl.edu/etj/articles/leapz.pdf $\endgroup$ – Robert Long Mar 3 '15 at 21:51

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