I am thinking about a problem where I need to estimate the maximum of a quadratic regression, that is, fit a model $y = ax^2 + bx + c + \epsilon$ and estimate the maximum as $-\hat b/(2\hat a)$. My question concerns the accuracy of the estimate as a function of the characteristics of the data. In particular, I envisage situations where most or all of the data are concentrated on one side of the maximum. I found this post, which shows how to calculate the variance of the estimate for a given data set.
What I want to do is to determine beforehand under what conditions such an estimate can be done with reasonable accuracy. For this purpose, I consider the following the model: The dependent variable follows a true unknown relationship $y = ax^2 + bx + c + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$. In addition, the independent variable $x$ is a random variable satisfying $x \sim N(\mu_x, \sigma_x)$. This situation is, of course, easy to simulate. However, I wonder whether one can say something analytically about the expected variance of the regression coefficients (which will allow to calculate an expected variance of the estimate of the maximum) as a function of sample size and the parameters $\sigma_\epsilon$, $\mu_x$ and $\sigma_x$.
Any help is greatly appreciated.
Michael.
