In a recent multiple choice examination I encountered a "select all that apply" type question, which had this statement among others:
If a sufficient estimator exists, it is always unique.
Now, I treated this statement as 'false' in the rush of the hall, knowing that there can be infinitely many sufficient statistics for a family. Back home, I tried to clarify the problem as I deemed fit, which read:
Disprove or prove: For a family $f(x,\theta), \theta\in\Theta$, if T(X) and U(X) be two sufficient unbiased estimators for the parameter $g(\theta)$, then $T=U$ almost surely.
I thought disproving it might be a trivial task, as it doesn't say anything about completeness (which would imply UMVUE by Lehmann Scheffe), only sufficiency. But I was stumped-- in the common families of distributions, the different sufficient statistics (like taking one-one functions of the natural sufficient estimators) all gave different expectations.
Could anyone please lend me some helpful hints towards disproving (or proving) this statement (Or if my rewording is not accurate)? Thanks a lot!
Any statistic of the form $f(T(X))$ for a bijective function $f(u)$ is a sufficient statistic. For it to be an unbiased estimator we need: $$E(f(T(X)) \ | \ \theta) = E(T(X) \ | \ \theta) \implies E(f(U) \ | \ \theta) = E(U \ | \ \theta)$$
Let $U = T(X)$ and $P_U(u|\theta)$ be pdf of $U$. $$S_{r,\theta} = \{u: P_U(u|\theta) = r \}$$ then $$M_{r} = \cap_{\theta \in \Theta}S_{r,\theta}$$
Any appropriate function $f$ such that $F_r = \int_{M_r} f(x) dx$, $T_r = \int_{M_r} x dx$, $F_r = T_r$ will be such that $f(T(X)) \neq T(X)$ but $E(f(T(X))) = E(T(X))$.
We can change the criteria to $f$ such that, $$r F_r + r' F_{r'} = r T_r + r' T_{r'}$$
Hence as long as there a non-trivial set $M_r$, we can always find two non-trivial sufficient stats which are unbiased estimates.
Put in the comments if you find a non-trivial $M_r$ for some $r$.