Complete and Sufficient Statistics

Question

Question: Let $X_1, X_2, ... , X_n$ be a random sample from a Poisson distribution with parameter $\theta$. Show that $Y = $$\sum_{i=1}^n X_i$ is a complete and sufficient statistic for $\theta$ .

I used factorization theorem to prove that $Y = $$\sum_{i=1}^n X_i$ is a a sufficient statistic. By definition,$Y = $$\sum_{i=1}^n X_i$ is complete if $E_\theta$$\sum_{i=1}^n X_i=0$ for all $\theta$. However, I just don't know where to start to show that it is complete.

Any suggestion or step-by-step answer is really appreciated. Thanks all!

Answer 1

For simplicity, denote $\sum_{i = 1}^n X_i$ by $T$. By assumption, $T \sim \text{Poisson}(n\theta)$. Thus for any measurable function $g: \mathbb{R}^1 \to \mathbb{R}^1$ such that $E_\theta[g(T)] = 0$, we have $$\sum_{k = 0}^\infty g(k) e^{-n\theta}\frac{(n\theta)^k}{k!} = 0, \quad \forall \theta > 0,$$ which implies $$f(\theta) \equiv \sum_{k = 0}^\infty \frac{g(k)n^k}{k!}\theta^k = 0, \quad \theta \in (0, \infty). \tag{1}$$

Extend the domain of $f(\theta)$ to $\mathbb{R}^1$ so that it can be viewed as a power series. $(1)$ says that $f(\theta)$ converges at every point in $(0, \infty)$, which further implies $f(\theta)$ converges everywhere in $\mathbb{R}^1$, for the convergence interval of a power series is always an interval that is symmetric about $0$. It then follows that $0 \equiv f^{(k)}(0) = g(k)n^k, k = 0, 1, 2, \ldots$ (see, for example, Corollary 8.10 of Rudin's Principles of Mathematical Analysis). From here to completeness of $T$ is immediate.