Suppose Jim observes and wishes to make inference on . Having taken a course in mathematical statistics, Jim identifies a minimal sufficient statistic , calculates it, and throws the rest of the data away.
Later, Jim realizes he actually had another observation 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 is minimal sufficient for , both unknown, and given a new observation it is trivial to calculate a new minimal sufficient statistic.