Exercise: Let $\phi$ be a positive (Borel) function on $\mathbb{R}$ such that $\int\limits_{a}^b\phi(x)dx < \infty$ for a pair $\theta = (a,b)$, with $-\infty<a<b<\infty$. Let $\Omega = \{\theta \in \mathbb{R^2}:a<b\}$. Define $$f(x|\theta) = c(\theta)\phi(x)\mathbb{1}{(a,b)},$$ with $c(\theta)$ such that $\int f(x|\theta)dx = 1$. Then $\{f(\cdot|\theta), \theta\in\Omega\}$ is called a truncation family. Suppose that $X_1,...,X_n \stackrel{iid}{\sim} f(\cdot|\theta)$. Let $X = (X_1,...,X_n)$. Show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic.
Question: How do I show that $T(X)$ is a sufficient statistic?
I know that according to the mathematical definition $T(X)$ is sufficient if the distribution of $X$ given $T$ is known (does not depend on $\theta$). However, I think it's quite hard to find the conditional distribution and show that it's not dependent on $\theta$.
I also know that $T(X)$ is sufficient if no other statistic that can be calculated from the same sample provides more information regarding $\theta$. So I need to show that $T(X)$ gives us just as much about $\theta$ as $S(X) = (X_1,...,X_n)$ does. Intuitively I feel this can be done by looking at $f(x_1|\theta)$ and $f(x_2|\theta)$ and then inspecting what happens with $\mathbb{1}_{(a,b)}(x)$, but I'm not sure how.
Thanks in advance!
Just use the Factorization Criteria, i.e., $$ \mathcal{L}(\theta ; x_1, ..., x_n) = c^n(\theta)I\{x_{(1)}\ge a\}I\{x_{(n)}\le b\}\prod_{i=1}^n\phi(x_i) , $$ thus $(x_{(1)}, x_{(n)}) $ is the sufficient statistic for $\theta = (a, b)$.