Are These Conjectures Regarding Sufficient Statistics True? I have these conjectures that I cannot quite prove (unless I impose another regularity condition of parameter-independent support for distribution, in which case, the conjectures are trivially true --- I think).
Would appreciate 1) confirmation (w/ proof ideally), or 2) rejection (w/ counter example ideally).
Conjectures:
Suppose $X_1, ..., X_n$ are IID scalar random variables with PDF $f(x; \theta_1, ..., \theta_k)$, where $k < n$ and $\theta_1, ..., \theta_k$ are well-identified without any constraints among them (I can make this technically precise if necessary). Suppose there are $s$ statistics, $T_1, ..., T_s$, each a scalar, that are jointly sufficient.  Then,
A) $s \ge k$;
B) If $s = k$, then the sufficient statistics are minimal;
C) If $s = k$, then the sufficient statistics are complete.
End of Conjectures
Notes: If I further assume that the support of $f(x)$ is independent of $(\theta_1, ..., \theta_k)$, it is my understanding that there is a theorem that shows that $f(x)$ must be from the exponential family and all the conjectures above are true.

$U[0,\theta]$ is a counterexample to C. $T_1=\max_i X_i$ is sufficient for $\theta$ but it is not complete because $$E\left[\frac{n+1}{n}T_1\right]=\theta=E[2\bar X]$$
(I believe A and B are true.)
