This likely isn't a research-level question, but it is at least a question of interest to this researcher. I'm happy with an answer that sends me somewhere (preferably online) to read about a well-known (to somebody) solution, if there is such a thing. First the question, then the motivation.

Let $X_1,X_2,\dots,X_N$ be independent $N(\mu,\sigma^2)$ random variables ($\mu$ and $\sigma^2$ are unknown), and let $Y_1,Y_2,\dots,Y_N$ be defined by $\{X_1,\dots,X_n\}=\{Y_1,\dots,Y_N\}$ and $Y_1 < Y_2 < \cdots < Y_N$. For $k < N$, how can one estimate the parameters $\mu,\sigma^2$ from $Y_1,Y_2,\dots,Y_k$?

The application that I need this for is a strategy to buy light bulbs. I just purchased a stunning chandelier with 27 tiny light bulbs. The bulbs themselves are strange, and I'll have to special-order them. After the first several burn out, I'd like to be able to estimate how many will burn out in, say, the next 6 months. Although the bulbs are independent, the lifetimes of the first 3 to burn out are not.

This problem must come up all the time: medical researchers don't know how long their patients will last, they only know how long some of them didn't.