Solution to Problem 3.

(3) [15 points] Let $S$ and $T$ be statistics, and denote the unknown parameter by $\theta$. Assume the statistical model has either a pdf or pmf for the observable. Let $h$ be any reasonable (i.e., measurable) function. For each of the following statements, either give a proof or a counterexample. (a) If $T$ is sufficient for $\theta$ and $T = h(S)$ for some function $S$, then $S$ is sufficient for $\theta$.

Solution: Using the factorization theorem, we know for some functions $g$ and $v$

$$f(x\vert\theta) \; = \; g(T(x)\vert\theta)v(x) \; = \; g(... ...))\vert\theta) v(x) \; = \; \tilde{g}(S(x)\vert\theta) v(x) ,$$

so by the factorization theorem, $S(x)$ is sufficient. The statement is True.


(b) If $T$ is complete for $\theta$ and $S = h(T)$, then $S$ is complete.

Solution: Completeness of $T$ means for any function $\phi$, $E_{\theta} [ \phi(T) ] = 0$ for all $\theta$ implies $P_{\theta} [ \phi(T) = 0 ] = 1$ for all $\theta$. If $S = h(T)$, then $\psi(S)$ $=$ $\psi(h(T))$ $=$ $\phi(T)$, so $E_{\theta} [\psi(S)] = 0$ for all $\theta$ means $E_{\theta} [ \psi(h(T)) ] = 0$ for all $\theta$, which means $P[ \psi(h(T)) = \psi(S) = 0 ] = 1$ for all $\theta$, so $S$ is complete. The statement is True.


(c) If $T$ is ancillary for $\theta$ and $T = h(S)$, then $S$ is ancillary.

Solution: Ancillarity means the distribution of $T$ doesn't depend on $\theta$. Of course, if $T$ is a statistic and $X$ is the data, then $T = h(X)$, but the distribution of $X$ does depend on $\theta$. Hence, the statement is False.