Does Delta Method work for asymptotic distributions?

[FallenApple]

So if I have a logistic regression: $log (\hat {odds})=\hat{\beta_{0}}+\hat{\beta_{1}}x$. How would I find a confidence interval for x if I am given $odds=5$ This is going in reverse, where if I have the outcome, I try to do inference on the predictor.

We know that $\hat{\vec{\beta}}$, the vector of $\hat{\beta_{0}}$ and $\hat{\beta_{1}}$, is distributed asmptotically as $N( \vec{\beta}, \hat{{Var{\hat{\beta}}}})$ for large sample sizes where $\hat{{Var{\hat{\beta}}}}$ is the estimated variance covariance matrix.( similar to how $\sigma^{2}$ is estimated by $s^{2}$ for the basic t test).

So solving for x using $log (5)=\beta_{0}+\beta_{1}x$ I get: $x( \beta_{0},\beta_{1}) =\frac{ log(5)-\beta_{0}} {\beta_{1} }$ a multivariate function of both parameters.

So it seems that I would use the delta method but the one problem is the variance covariance matrix is an estimated one, so it doesn't have any actual parameters in it. Would delta method still work for this? Or would I need to try another method?

 

[DrDu]

No, that's ok. The point is that you can Taylor expand $Var_\beta=Var_\hat{\beta}+(\partial Var_\beta/\partial \beta )(\hat{\beta}-\beta)+\ldots$.
Now $\hat{\beta}-\beta$ is of order $1/\sqrt{N}$, so that asymptotically, you can neglect the second term.