Are the parameters of Non-linear regression independent of each other? I'm propagating error in the parameters determined by the following growth function...
$$
\hat{y} = ae^\frac{t}{b} + (1- a)e^\frac{t}{c}
$$
Say I have another model that uses the parameters {a,b,c} to estimate another rate $z$. 
$$
z = a \times b \times c
$$
I want to know the 95% confidence interval for z. Can I just assume the true parameter for $z$ falls between the product of the 2.5% limit for {a,b,c} and the product of 97.5% for {a,b,c}?
 A: The product of the confidence intervals for $a,b,c$ will not be a valid confidence interval for $z$, even in the case of $a,b,c$ being independent (which, for the record, is especially unlikely). As a simple example, suppose we only considered $a$ and $b$ and the confidence interval for $a \times b$. If the each of these confidence intervals were (-1,1), then using that formula, we would end up with (1,1) as our confidence interval for $a \times b$. If $a$ and $b$ were independent, this would clearly be false. 
If you have the covariance matrix for $a, b, c$, and a fairly large sample size, a straightforward method is to use the delta method (https://en.wikipedia.org/wiki/Delta_method). In general, if $\beta$ is your vector of parameters, if you'd like a confidence interval of $f(\beta)$, it can be created from 
$f(\hat \beta) \pm z * ( f'(\hat \beta)^T * \hat \Sigma * f'(\hat \beta))$
where $f$ is your function of interest (in this case, we can think of $f(\beta) = \beta_1 \times \beta_2 \times \beta_3$ where $\beta_1 = a, \beta_2 = b, \beta_3 = c$), $f'$ is the vector of derivatives and $\hat \Sigma$ is the estimated covariance matrix and $z$ is the standard z-score associated with the confidence interval. 
