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Suppose $x_i$ are drawn i.i.d. from a $p$-variate Gaussian, $\mathcal{N}\left(\mu,\Sigma\right)$. Suppose one observes $x_1,x_2,\ldots,x_n$. One also observes $s_{n+1},s_{n+2},\ldots,s_{n+m},$ where $s_i = \mbox{sign}\left(x_i\right)$ is a $p$-vector consisting of -1 and +1's. (Well, in principle, it could contain some zeroes...)

I would like to estimate $\mu$ from this information. One should be able to do at least as well as the estimate that ignores the $s_i$ (namely $\hat{\mu} = (1/n) \sum_{1\le i \le n} x_i$). But is there a method that does better? How good is the method? Is this a well-known problem?

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Last m observations you can consider as censored. You could update the censored observation likelihood by probability of region.

For Example let p=3. you got the sign vector (1,-1,-1), the corresponding representation in likelihood will be $P[X_1>0,X_2<0,X<0]$.

Adding extra information is always better. This problem cannot solve analytically. Solving this problem involve numerical optimisation.

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  • $\begingroup$ I interpret this answer as: compute the MLE of $\mu$ based on the $x_i$ in the 'usual way', and use the sign information based on a sign-censored Gaussian likelihood. And the whole thing will have to be done numerically. Is that right? $\endgroup$ – shabbychef Mar 27 '12 at 17:49
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    $\begingroup$ You are right. Use the censor information in the ML estimation. $\endgroup$ – vinux Mar 27 '12 at 18:03

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