(1)part(a): can minimal sufficient be independent of observation?
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let $X$ is one observation from $ U(\theta, \theta+1)$, $\theta \in \{0,\pm1 ,\pm 2 ,\cdots \}$
$T=\lfloor X \rfloor$ is minimal sufficient for $\theta$.
for any $\theta_0\in \{0,\pm1 ,\pm 2 ,\cdots \}$
$\frac{f_{\theta}(x)}{f_{\theta_0}(x)}=\frac{\theta_0}{\theta}\frac{1_{(\theta < X < \theta+1)}}{1_{(\theta_0 < X < \theta_0+1)}}$
since $\theta$ is integer
$=\frac{\theta_0}{\theta} \frac{1_{(\lfloor X \rfloor =\theta)}}{1_{(\lfloor X \rfloor =\theta_0)}}$ depend on $X$ through $T=\lfloor X \rfloor$
so $T$ is a minimal sufficient.
$P_\theta(\{X\leq y\} \cap \{\lfloor X \rfloor=t\})=\left\{ \begin{array}{cc} 1 & y>t=\theta \\ y & if \lfloor y \rfloor =t=\theta \\ 0 & O.W \end{array} \right.$
$P_\theta(\{X\leq y\}) \times P_\theta(\{\lfloor X \rfloor=t\})= \left\{ \begin{array}{cc} 1 & y>t=\theta \\ y & if \lfloor y \rfloor =t=\theta \\ 0 & O.W \end{array} \right.$
so $X$ and $\lfloor X \rfloor$ are independent. Am i wrong?
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part(b) when we have $n$ observation
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let $X_1, \cdots , X_n$ are i.i.d random variable from $ U(\theta, \theta+1)$, $\theta \in \{0,\pm1 ,\pm 2 ,\cdots \}$
(2)find the minimal sufficient:
$\lfloor X_1 \rfloor=\lfloor X_2 \rfloor=\cdots = \lfloor X_n\rfloor=\theta=\lfloor \max\{X_i\} \rfloor=\lfloor \min\{X_i\} \rfloor$
all of them are minimal sufficient?
(3) is $X_1$ sufficient? (when we have $n$ observation.
since if not, a function of it, $\lfloor X_1 \rfloor$ is sufficient! I am confused!
(main question (1) (2) (3))