Sufficient statistic based on a random sample of size n where the pmf is $P_{\theta}(X=x) = c(\theta)2^{-x/\theta}$

Question

so the question asks to find a sufficient statistic based on a sample of size n, where $X \sim f_{\theta}(x)$ and \begin{equation} f_{\theta}(x) = P_{\theta}(X=x) = c(\theta) 2^{-x/\theta}, \quad x = \theta, \theta+1,..., \theta >0 \end{equation} and \begin{equation} c(\theta) = 2^{1-1/\theta}(2^{1/\theta}-1) \end{equation} I derived that the joint pmf is \begin{equation} P_{\theta}(X_1 = x_1,...,X_n = x_n) = c(\theta)^{n}2^{-1/\theta \sum x_{i}}\mathbb{I}_{\{X_{(1)}\geqslant \theta\}} \end{equation} My question is, in this case, is a sufficient statistic $T(x) = (\sum x_{i}, X_{(1)})$ or both $\sum x_{i}$ and $X_{(1)}$ can serve as a sufficient statistic? Thanks so much for your time and consideration.

Answer 1

You write that the question asks to find “the sufficient statistic”, but there is never only one sufficient statistic – in particular, any non-zero multiple of a sufficient statistic is also a sufficient statistic. The question probably meant to ask for “a sufficient statistic”.

The statistic you found is indeed sufficient. Neither of its components is sufficient. By definition, if a statistic is sufficient, the likelihood of the data depends on the data only through the statistic. By contrast, different data can lead to different $\sum_iX_i$ (and thus a different likelihood) but the same $X_{(1)}$, and vice versa. Thus neither of these components is sufficient by itself.