# Regression and variable with errors modelling problem

Wikipedia says that almost all Measurment error models can be formulated as follows:

Usually measurement error models are described using the latent variable model latent variables approach. If ''y'' is the response variable and ''x'' are observed values of the regressors, then we assume there exist some latent variables ''y*'' and ''x*'' which follow the model's "true" functional relationship ''g'', and such that the observed quantities are their noisy observations:

$\begin{cases} x = x^* + \eta, \\ y = y^* + \varepsilon, \\ y^* = g(x^*\!,w\,|\,\theta), \end{cases}$

where $\theta$ is the model's parameter and ''w'' are those regressors which are assumed to be error-free (for example when linear regression contains an intercept, the regressor which corresponds to the constant certainly has no "measurement errors").

The variables ''y'', ''x'', ''w'' are all ''observed'', meaning that the statistician possesses a data set of ''n'' statistical units $y_i$, $x_i$, $w_i$ which follow the data collection|data generating process described above; the latent variables ''$x^*$'', ''$y^*$'', ''$\epsilon$'', and ''$\eta$'' are not observed however.

I am dealing with a particular problem that arises from measuring an electronic device for which I can make a similar model. The difference is that I don't observe $y$, but I know $h$ and $g$ and I can observe different $x \in \mathbb{R^p}$ for which hold:

$$\begin{cases} x = x^* + \eta, \\ h(x^*) = g(x^*\!,w\,|\,\theta)\\ \end{cases}$$

I want to name this problem so I can properly study it. Is it still some kind of Variable-with-error-model or is it something different?

The actual formulation of my problem is: $$\begin{cases} x = x^* + \eta, \\ h(x^*) = f(x^*)\theta_1+\theta_2\\ \end{cases}$$

I observe $x_i$ i know that the equation holds and i want to make inference about $\theta_1$ and $\theta_2$

• The model appears self-contradictory. Written more explicitly, it makes an assertion about the random variable $\epsilon$: $f_1(x-\epsilon)=\beta f_2(x-\epsilon)+\alpha$. In general, given $f_1$ and $f_2$, there will not exist any constants $\alpha$ and $\beta$ that can make this true as statement about random variables unless $f_1$ and $f_2$ already enjoy such a functional relationship for all real numbers--in which case you scarcely need to make any observations at all, because you can just solve for $\alpha$ and $\beta$! What am I misunderstanding?
– whuber
Jul 30, 2013 at 18:20
• the equality only holds for $x^*$. not for any $x$. I will try to re-write the question so it's clearer Jul 30, 2013 at 18:23
• You appear to be treating the $x^*$ as unknown constants, so at a minimum your model must apply to some range of them. But the defining equation $f_1(x^*) = \beta f_2(x^*) + \alpha$ determines $\beta$ and $\alpha$ uniquely once you substitute two separate values for $x^*$ into that equation, assuming $f_1$ and $f_2$ are not linearly related, and will have no solutions upon substituting three values of $x^*$. Could you perhaps tell us something about your data and the experimental setting rather than proposing a (mysterious) model? Perhaps that would clear things up.
– whuber
Jul 30, 2013 at 19:54
• I understand what you say, but i am not claiming that $f_1=\beta f_2+\alpha$. This equality only holds for some values of their domain ($x_i^*$), which are the ones i can observe with error Jul 31, 2013 at 15:57
• i erased the previous question, in order to give a more precise (altough more general) question Jul 31, 2013 at 16:04