Let $(X_1,X_2,..,X_n)$ be a random sample following a $Be(1,\theta)$. Is the $T=X_1-X_2$ sufficient? Is it complete?
Completeness:
If I assumed $T$ to be sufficient. I know by the factorization lemma that $P(x)=f(x|\theta)=g(t|\theta)+h(x)$. Therefore I could consider $h:(x_1,x_2)\to x_1-x_2$ so that $E(h(X_1,X_2))=0$ however $P(X_1-X_2=)\neq 1$ which means that $T$ is not complete.
Sufficiency:
I am having problems proving sufficiency:
I though of using the Factorization lemma, and factorize $f(x|\theta)$, that would yield:
$f(x|\theta)=(\theta^{x_1}+(1-\theta)^{1-x_1}))(\theta^{x_2}+(1-\theta)^{1-x_2})$
However after many attempted algebraic manipulation including the use of logarithm, I cannot separate the thetas from x.
Question:
How should I prove sufficiency of $T$?
Thanks in advance!
As @StubbornAtom suggested, find $\mathbb P(X_1=k_1,\ldots,X_n=k_n\mid X_1-X_2=0)$. Does this probability depend on $\theta$ or does not? This is more than sufficient to answer if $X_1-X_2$ is sufficient or not.
In addition, few words on concept of sufficiency. Sufficient statistics are those whose knowledge carries the same information about an unknown parameter as knowledge of the entire sample. Can the knowledge of the difference of the first two elements of the sample replace the knowledge of the entire sample? Why then do people toss a coin many times, trying to estimate success probability? They would have thrown it twice and, based on the difference in the results of the first two throws, made a conclusion about the probability of success.