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.