Question: Let $r_t \sim \mathcal{N}(\mu, \sigma^2)$ and $p = P(r_t \leq -r_T) = F(-r_T)$. Suppose that $p=.05$, and we want to find a consistent estimator for $r_T$.
So far, I have that
$$\begin{align*} .05 &= P\left(\frac{r_t - \mu}{\sigma} \leq \frac{-r_T - \mu}{\sigma}\right) \\ &= P\left(Z \leq \frac{-r_T - \mu}{\sigma}\right) \\ &= \Phi\left(\frac{-r_T - \mu}{\sigma}\right) \\ &= 1 - \Phi\left(\frac{r_T + \mu}{\sigma}\right) \end{align*}$$
How do we decide on a consistent estimator given $\Phi\left(\frac{r_T + \mu}{\sigma}\right)$? The sample mean of $r_t$ seems like a decent candidate, since that is a consistent estimator (because $r_t$ is normal). However, don't we need to take .05 into consideration somehow?
If it helps any, this question is within the domain of financial economics, where $r_t$ is log-returns of a specific asset.
Using your derivations, $r_T=-\mu-\Phi^{-1}(0.05)\sigma$. Thus, $$ \hat{r}_T=-\hat{\mu}-\Phi^{-1}(0.05)\hat{\sigma} $$ is consistent if $\hat{\mu}$ and $\hat{\sigma}$ are consistent estimators of $\mu$ and $\sigma$.