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:
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?