A lot of the time, the purpose of conducting our experiments is to find the value of θ and to find the bigger picture. A sufficient statistic is one that gives us all the information we want about θ.
Lucky for us, the factorization theorem gives us the ability to find sufficient statistics. It states that T is sufficient for θ if we have the following:
Here, g and h are arbitrary functions.
In general, if T is a sufficient statistic, then it does not lose any information about θ and the best statistic is the one that gives us the maximal reduction. We call this the minimal sufficient statistic; in its definition, T(X) is minimal ...
