Estimate normal distribution parameters from smallest N samples I have a bunch of small datasets (billions of sets of 7 samples). Each dataset represents the smallest 7 samples of a larger set of 15 values which are normally distributed. Given just the smallest 7 samples, how could I calculate the mean/stdev for the full 15 sample set efficiently?
I suspect this involves using order statistics and the beta distribution but this is going beyond my usual depth in stats.

The joint density is given at Joint density of first r order statistics, and then just use maximum likelihood. Leaving out the combinatorial factor (which is irrelevant to the maximization), the log likelihood is (with $f$ denoting the normal density function, $F$ the corresponding cdf)
$$ \ell(\mu,\sigma)=\sum_{i=1}^7 \log f(x_i) + (15-7)\log\left(1-F(x_7)\right) $$
and then just use numerical optimization.
For more details see the answer at the similar question https://stats.stackexchange.com/a/276322/11887
