A statistic is said to be sufficient for if the conditional distribution , that is, doesn’t depend on . Intuitively, we mean that a given statistic provides as much information as possible about ; there isn’t a way to “crunch” the observations in a way to yield more data.
Of course, this is equivalent to the likelihood function factoring into a function of and alone and a function of the alone.
Though the problems can be assigned to get the students to practice using the likelihood function factorization method, I think it is important to provide an example which easily shows what sort of statistic would NOT be sufficient for a parameter.
Here is one example that I found useful:
let come from a uniform distribution on .
Now ask the class: is there any way that could be sufficient for ? It is easy to see that will converge to 0 as goes to infinity.
It is also easy to see that the likelihood function is where is the standard Heavyside function on the interval (equal to one on the support set and zero elsewhere) and is the of maximum magnitude (or the order statistic for the absolute values of the observations).
So one can easily see an example of a sufficient statistic as well.