Let the sample $x=\left(x_{1}, \ldots, x_{n}\right)$ have likelihood function $f(x ; \theta)$. What does it mean to say $T(x)$ is a sufficient statistic for $\theta$ ?

Show that if a certain factorization criterion is satisfied then $T$ is sufficient for $\theta$.

Suppose that $T$ is sufficient for $\theta$ and there exist two samples, $x$ and $y$, for which $T(x) \neq T(y)$ and $f(x ; \theta) / f(y ; \theta)$ does not depend on $\theta$. Let

$$T_{1}(z)= \begin{cases}T(z) & z \neq y \\ T(x) & z=y\end{cases}$$

Show that $T_{1}$ is also sufficient for $\theta$.

Explain why $T$ is not minimally sufficient for $\theta$.