Do you know some good references (papers or books) for the theoretical inference in Deming's regression model ?
EDIT: I was a little disconcerted about a point in Ripley and Thompson's paper Regression techniques for the detection of analytical bias. I size the opportunity to share my clarifications (thanks to @Procrastinator for his help)
The postulated model is $x_i = u_i + \delta_i$ and $y_i = v_i + \epsilon_i$ where $u_i$ and $v_i=\alpha+\beta u_i$ are fixed numbers and the error terms $\delta_i$ and $\epsilon_i$ are independent. This is the commonly called functional relationship model. See Cheng & Ness's paper "Structural and functional models revisited" for a thorough review of the functional relationship model. But this is also nothing but Deming's well-known regression model.
After substituting $x_i-\delta_i$ for $u_i$ in the expression of $y_i$ one gets $y_i=\alpha+\beta x_i + (\epsilon_i - \beta \delta_i)$ and Ripley and Thompson consider $y_i=\alpha+\beta x_i + (\epsilon_i - \beta \delta_i)$ as a weighted least squares regression model with covariate $x_i$ and error term $\epsilon_i - \beta \delta_i$. The sentence on page 349 "... showing dependence of $y_i$ on $x_i$" is quite disconcerting: $x_i$ and $y_i$ are independent in the original postulated model. In fact the authors do not consider that $y_i=\alpha+\beta x_i + (\epsilon_i - \beta \delta_i)$ provides a valid weighted least squares model but they do not explain why this model is not valid. The cause, which is not clearly mentioned in the paper, is that $x_i$ is not independent of the error term $\epsilon_i - \beta \delta_i$ (I guess there are more details in Ripley's unplublished paper mentioned in the references).
In fact one of the purposes of this paper is to discuss about the validity of "approximately valid" models.
I'm still looking for a reference with a thorough presentation of the functional model and related models (structural and ultrastructural). Cheng & Ness's paper "Structural and functional models revisited" is a nice rigorous review about the theoretical treatment of the model but it does not address the more practical questions.
