Sufficient statistic for a function of the parameter

Question

We know that if $T$ is a sufficient statistic for $\theta$ then $f(T)$ is a sufficient statistic for $f(\theta)$ if $f(.)$ is a one -one function. But,what if $f$ is not one one? For example, in case of Bernoulli $(p)$ ,how to find the sufficient statistic for $p(1-p)$?

Answer 1

If $T$ is a sufficient statistic for $\theta$, then

• $T$ is a sufficient statistic for $f(\theta)$ for any function $f$,
• $f(T)$ is a sufficient statistic for $\theta$ for any injective function $f$.

Since you want a sufficient statistic for $f(\theta)$, this function doesn’t need to be injective.