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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.

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While this does not answer the problem as you've formalized it, I would argue that the correct method to consider this problem is actually Weibull analysis--generally failure times are not distributed normally. Essentially, plot the failure times on Weibull paper; the slope of the line you get will give you the information you want. –  Daniel Litt Mar 3 '11 at 18:15
    
Do you really expect the lifetime of a light bulb to be normally distributed? –  Ori Gurel-Gurevich Mar 3 '11 at 18:31
    
I went with normal for two reasons: if it isn't normal it likely is close, and second, if I can't do this problem with normal distributions I'm unlikely to be able to handle anything else. –  Kevin O'Bryant Mar 3 '11 at 20:47
    
Daniel's comment seems to be the answer! This link weibull.com/AccelTestWeb/… has all the detail I need. Daniel, put your comment in an answer, please, so I can accept it. –  Kevin O'Bryant Mar 3 '11 at 20:51
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"I just purchased a stunning chandelier with 27 tiny light bulbs."?! Come on man a lot of us are grad students here. –  rcompton Mar 4 '11 at 2:20
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up vote 3 down vote accepted

As requested, I'm making my comment an answer.

While this does not answer the problem as you've formalized it, I would argue that the correct method to consider this problem is actually Weibull analysis--generally failure times are not distributed normally. Essentially, plot the failure times on Weibull paper; the slope of the line you get will give you the information you want.

As Kevin O'Bryant has pointed out, this site does Weibull computations.

For a beautiful theorem explaining the relevance of the Weibull distribution, see the Fisher-Tippett-Gnedenko theorem. The model to keep in mind is the following: objects generally have more than one possible point of failure, and failure occurs when the first point of failure breaks. So in an ideal case one wants to compute the minimum of a collection of IID's. The Fisher-Tippett-Gnedenko theorem gives that the extreme value distributions, of which the Weibull distribution is an instance, are universal for this situation, just as the central limit theorem gives that the normal distribution is universal for the mean of a family of IID's.

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