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Suppose $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ and we only observe $(X_1 - \bar{X}, \ldots, X_n - \bar{X})$, such that each observed point is now mean centered. Can we still learn about $\mu$ or $\sigma^2$?

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Since the variables are de-meaned, I don't think there is a way for you to get back the mean. However, you can make inferences about the variance since (by the way I'm assuming your samples are uncorrelated):

\begin{align} \operatorname{var}(X_i-\bar{X})&=\operatorname{var}\left(\frac{n-1}{n}X_i-\frac{1}{n}\sum_{i\neq j} X_j\right) \\&=\frac{(n-1)^2}{n^2} \operatorname{var}(X_i) + \frac{1}{n^2} \sum_{i\neq j} \operatorname{var}(X_j) \\&=\sigma^2\frac{(n-1)^2}{n^2}+\sigma^2\frac{(n-1)}{n^2}\\ &=\frac{n-1}{n}\sigma^2 \end{align}

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    $\begingroup$ +1. One rigorous way to support your statement about the mean being unidentifiable would be to observe that $\mu$ doesn't enter into the likelihood at all. $\endgroup$ – whuber Apr 4 at 13:56

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