I am working on confidence intervals for transformed parameters in dose-response log-logistic model.
For simplicity, let's assume 2 parameter regression model with normal errors, where $\theta=(a,b)$ are the parameters:
$$E[y|x] = \frac{1}{1 + \exp(a(\log x -\log b))}$$
Now, I am interested in finding the confidence interval for $ED_c= g(a,b|c) = \exp{(\frac{c}{a}}-\log b)$, where $c$ is a user-chosen constant.
I know that you can do monotone transformations of confidence intervals for 1-dimensional parameters, e.g., a log transformation of the parameter and both boundaries of the confidence interval.
However, I am not sure if the same can be applied for the transformation I am interested in, since it's a $R^2\rightarrow R$ function and the notion of monotonicity in multiple dimensions kinda vanishes.
Now, I can construct the confidence region (i.e. ellipse) for parameters $a,b$ via LRT test quite easily, let's call this confidence region $C$. My idea is to evaluate my function $g$ on the set $C$ and by finding the minimum and maximum of $g$ on the set $C$ I hope for acquiring the confidence interval for the $ED_c$.
I think it boils down to just implication $\theta \in C \implies g(\theta) \in g(C)$, but I might be missing something.
My intuition is that it should work well, however I struggle to find theoretical justification.
Disclaimer: I know about bootstrapping, Wald type, reparameterization, etc., I am specifically interested in this particular construction.