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From a Physics equation I have the following model:

$$W=\beta_0+\beta_1Z$$

$\beta_0$, $\beta_1$ are fixed values for which I want to find a $1-\alpha$ confidence region.

I have $(z_1,w_1),(z_2,w_2),\ldots,(z_n,w_n)$ observations.

I get these observations from evaluating the function $f: \mathbf{R}^{l+k+2}\rightarrow \mathbf{R}^2$.

$$f\left(X_1,\ldots,X_l,Y_{1i},\ldots, Y_{ki},P_{1i},P_{2i}\right)=\left(f_1\left(\ldots\right),f_2\left(\ldots\right)\right)=(Z_i,W_i),$$

where $X_j\sim\mathcal{N}(\mu_j,\sigma^2)$ $Y_{ij}\sim\mathcal{N}(\theta_{ij},\sigma^2)$ $P_{ik}\sim\mathcal{N}(p_{ik},\frac{p_{ik}}{1000})$; $\sigma^2$ is known, but not $\mu_j,\theta_{ji},p_{ji}.$

$X,Y,P$ are measures from an electronic instrument.

The equation $w_i=\beta_0+\beta_1z_i$ holds when $\left(z_i,w_i\right)=f\left(\mu_1,\ldots,\mu_l,\theta_{1i},\ldots,\theta{ki},p_{1i},p_{2i}\right)$ ($\bf{not}$ when $\left(z_i,w_i\right)=\mathbf{E}(f(\ldots))$

Note that there are both common $\left(X_j\right)$ and not common variables $\left(Y_{ji} , P_{ki}\right)$ for all observations. Still It happens that $Y_{ji}=Y_{st}$ and $P_{1i}=P_{2t}$ for some $j$ $i$ $s$ and $t$. (Meaning that variables $Y$ and $P$ are used for more than one observation, but not all)

$f_1$ and $f_2$ are rational functions. Both numerators and denominators are quadratic functions when restricted to one variable. They are well defined.

So, what can i do to get confidence regions for $(\beta_0,\beta_1)$?

A regression model and Maximum Likelihood techniques are tempting.

\begin{equation} \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix} = \begin{pmatrix} 1 & z_1 \\ \vdots \\ 1 & z_n \end{pmatrix} \begin{pmatrix} \beta_0 \\ \beta_1 \end{pmatrix} + \begin{pmatrix} \epsilon_1 \\ \vdots \\ \epsilon_n \end{pmatrix} \end{equation}

$\bf{\epsilon}=\bf{W-\beta_1Z-\beta_0}$ (Is this right?)

As $(Z,W)$ has a complicated joint distribution, so does $\bf{\epsilon}$.

I can figure out what to do when each $(Z_i,W_i)\sim\mathcal{N_2}((z_i,w_i),\sum)$. But this is not the case.

I understand that there may not be an easy answer for these questions, but if anyone can give some general tips or point me to the right literature i can work from there.

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  • $\begingroup$ Is this the same question you asked at stats.stackexchange.com/questions/61376/…? If not, how does it differ? If it's the same, then please delete this one and edit your original. $\endgroup$ – whuber Jun 13 '13 at 14:12
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    $\begingroup$ Hi. I don't think is the same question. In my previous one i was asking how to model the problem. (For what I really aprecciate your help.) But in this one, I am asking how to find confidence interval with the model correctly stated. $\endgroup$ – Manuel Jun 13 '13 at 14:19
  • $\begingroup$ You can bootstrap. $\endgroup$ – Peter Flom Jun 13 '13 at 18:07
  • $\begingroup$ Thanks for the suggestion @Peter. I am reading about resampling residuals. It comes to me the question if independence of residuals is necesary for the method. $\endgroup$ – Manuel Jun 13 '13 at 18:59
  • $\begingroup$ Why wouldn't the residuals be independent? Is the longitudinal data? Clustered data? And, even if they are dependent there are probably ways to bootstrap it properly. $\endgroup$ – Peter Flom Jun 13 '13 at 19:41

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