Let $X_1,\dots X_n$ be an i.i.d sample from the distribution with frequency function $$P(X=x) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}$$ for $x = -1,0,1$.
- Use the factorization theorem to find a sufficient statistic for $\theta$.
- Does the distribution belong to the one-parameter exponential family.
I need help in these 2 questions. Im not sure how to apply the factorization theorem for question 1.
The conditional distribution of $(X_1,\ldots,X_n)$ given $|X_1|,\ldots,|X_n|$ does not depend on $\theta$, since $\Pr(X_1 = 1\mid |X_1|=1) = \Pr(X_1 = -1 \mid |X_1| = 1) = 1/2.$ Therefore $(|X_1|,\ldots,|X_n|)$ is a sufficient statistic, but it's not a minimal sufficient statistic. To see that, show that $|X_1|+\cdots+|X_n|\sim\operatorname{Binomial}(n,\theta).$
Note that $$ \prod_{k=1}^n \left(\frac \theta 2 \right)^{|x_k|} (1-\theta)^{1-|x_k|} = \left( \frac \theta 2\right)^{|x_1|+\cdots+|x_n|} (1-\theta)^{n - (|x_1|+\cdots+|x_n|)}. $$