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Consider the model: $$ y(\theta) = \alpha + \beta \cdot \tan(\theta + \gamma) + noise $$
where $\alpha$, $\beta$, and $\gamma$ are parameters I want to estimate and the noise is normally distributed: $N(0,\sigma)$. $\alpha$ and $\beta$ has the unit meters btw. The input variable is in the range: $$ -10\pi/180<=\theta<=10\pi/180 $$

Given a dataset consisting of N $(\theta_i,y_i)$ pairs I estimate the parameters by minimizing: $$ S = \Sigma_{i=0}^N [\alpha + \beta \cdot \tan(\theta_i + \gamma) - y_i]^2 $$

I ran a Monte Carlo simulation with $$ \alpha = 0.5 $$ $$ \beta = 1.0 $$ $$ \gamma = 4.0 \pi/180 $$ $$ \sigma = 0.0005 $$

The parameter estimates are nearly normally distributed: enter image description here

Why are the estimates for $\beta$ much better than those for $\alpha$ and $\gamma$?

And why are the estimates for $\alpha$ and $\gamma$ strongly correlated?

Andy
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    You can look at the sensitivity of the optimal solution with respect to the noise term. Or, you can use a different cost function to minimize. This is a well-studied problem and lot of things can be found regarding that on the Internet. – KBS Sep 30 '23 at 06:33
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    Something to keep in mind, your estimators for $\alpha$, $\beta$, and $\gamma$ will be correlated. – A rural reader Sep 30 '23 at 17:18
  • @Aruralreader: yes you are correct the estimators for $\alpha$ and $\gamma$ are strongly correlated. But how could you see that without performing any Monte Carlo simulation? – Andy Oct 02 '23 at 13:24
  • I doubt you’ll find a method that doesn’t involve computation. – A rural reader Oct 02 '23 at 14:50

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