If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ Also define the following statistic: $$ T_1(k) = \frac{1}{k} \sum_{i=1}^k X_i $$ Are the followings sufficient statistics?
(1) $T_1(k)$ when $k = n-c$, where $c$ is a constant integer.
(2) $T_1(k)$ when $k = c$, where $c$ is a constant integer.
Is the result here generalizeable to other distributions (with their proper statistics)?
$(1)$ and $(2)$ are the same statement, and $(2)$ is trivially not true---take $k = 1$ and $n > 1$.