Is the range of a sample of size n a "sufficient statistic"?

Question

I know that every order statistic themselves are sufficient statistic. The range is the max minus the minimum. Is the range also a sufficient statistic because it is a function of two sufficient statistics?

Answer 1

A few things.

1. Sufficient for what? A statistic is sufficient with respect to some parameters in a particular model. It makes no sense to say "the maximum is a sufficient statistic" without reference to a model. Nothing, other than the entire data set, forms a sufficient statistic without additional qualification as to what the model is.
2. The max is not generally a sufficient statistic, nor is the min. The order statistics collectively form a sufficient statistic whenever we have exchangeability (particularly in the case where the samples are i.i.d.) This just means that we can forget the order the samples came in and still have the same information about the parameters. (The max is sufficient for $\theta$ when the model is a uniform on $(0,\theta)$ but that is a very specifice situation.)
3. A function of sufficient statistics is not necessarily sufficient. Intuitively, the function must preserve all the information in the statistics. (They were already sufficient... putting them through a function can't help... it can only ruin it.) For instance, $f(x) = 0$ is a function. If $T$ is a sufficient statistic is $f(T) = 0$ sufficient? Of course not.