## Sufficiency Question

Suppose Jim observes $X_1, ..., X_m \stackrel{iid}{\sim} f(x | \theta)$ and wishes to make inference on $\theta$. Having taken a course in mathematical statistics, Jim identifies a minimal sufficient statistic $T(X_1, ..., X_m)$, calculates it, and throws the rest of the data away.

Later, Jim realizes he actually had another observation $X_{m + 1}$ that he didn't factor into the calculation of his minimal sufficient statistic. Can Jim, having thrown away most of his data, still recover a minimal sufficient statistic that takes into account all of the data?

This isn't a homework problem or anything, I'm just curious. Intuitively I would think the answer is yes, but I have no ideas for how to prove it (nor the time, since I have a bunch of midterms to study for ). An example of what I'm talking about would be that $(\sum X_i, \sum X_i ^ 2)$ is minimal sufficient for $X_i \stackrel{iid}{\sim} N(\theta, \sigma^2)$, both unknown, and given a new observation it is trivial to calculate a new minimal sufficient statistic.