I am wondering if it is possible to determine the covariance, $\text{Cov}(a,b)$, of two fitted parameters given I know their explicit relationship $a=a(b)$?
I would like to construct the covariance matrix in the space of the parameters $\left\{a,b\right\}$. Using the relationship $a=a(b)$, I can generate a table of, say, $n$ values for $a$ and $b$ and determine the covariance between $a$ and $b$ using $$\text{Cov}(a,b) = \frac{1}{n} \sum_{i=1}^n (a_i - \bar{a} ) ( b_i - \bar{b} )$$
It is not clear, however, how these $n$ values should be chosen so as to give the correct value of $\text{Cov}(a,b)$.
Another way to determine the standard deviation of, say, the best fitted $a$ is to plot the chi square vs. a and determine the a that gives $\Delta \chi^2 = 2.3$ (1 sigma deviation for a 2 parameter fit). Somehow this should agree with the result I get by taking the square root of $$\text{Cov}(a,a) = \frac{1}{n} \sum_{i=1}^n (a_i - \bar{a} )^2.$$
But, again, this depends on $n$.
So, given the relationship $a=a(b)$, how may I determine the covariance of these two parameters unambiguously?