Consider your data are $(Y_1,X_1)$, ... $(Y_{1000},X_{1000})$ iid and the following holds:
$$Y|X\sim Beta(X\beta, 1).$$
I want to estimate $\beta$. What is the function in R that gives me this estimate and its confidence intervals? I understand that using ML to obtain this $\beta$ is theoretically easy, but coding it is quite ugly, and I am sure there must be a function for this.
Assuming $Y_i \in (0, 1)$ and $X_i > 0$, the likelihood is $$L(\beta) = \prod_{i=1}^n X_i\beta Y_i^{X_i \beta - 1}$$ so the log-likelihood is $$\log L(\beta) = \sum_{i=1}^n (X_i \beta - 1) \log Y_i + \sum_{i=1}^n \log X_i + n \log \beta$$ Take the derivative with respect to $\beta$ and set it equal to zero. Solving the resulting equation for $\beta$ gives you a simple expression for the MLE.