Suppose that we have a linear model $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$ that meets all the standard regression (Gauss-Markov) assumptions. We are interested in $\theta = 1/\beta_1$.
Question 1: What assumptions are necessary for the distribution of $\hat{\theta}$ to be well defined? $\beta_1 \neq 0$ would be important---any others?
Question 2: Add the assumption that the errors follow a normal distribution. We know that, if $\hat{\beta}_1$ is the MLE and $g(\cdot)$ is a monotonic function, then $g\left(\hat{\beta}_1\right)$ is the MLE for $g(\beta_1)$. Is monotonicity only necessary in the neighborhood of $\beta_1$? In other words, is $\hat{\theta} = 1/\hat{\beta}$ the MLE? The continuous mapping theorem at least tells us that this parameter is consistent.
Question 3: Are both the Delta Method and the bootstrap both appropriate means for finding the distribution of $\hat{\theta}$?
Question 4: How do these answer changes for the parameter $\gamma = \beta_0 / \beta_1$?
Aside: We might consider rearranging the problem to give $$\begin{align*} x_i &= \frac{\beta_0}{\beta_1} + \frac{1}{\beta_1} y_i + \frac{1}{\beta_1} \epsilon_i \\ &= \gamma + \theta y_i + \frac{1}{\beta_1} \epsilon_i \end{align*}$$ to estimate the parameters directly. This doesn't seem to work to me as the Gauss-Markov assumptions no longer make sense here; we can't talk about $\text{E}[\epsilon \mid y]$, for example. Is this interpretation correct?