Complete statistic

Remarks

Every complete statistic is boundedly complete. Example: let $T$ have the binomial law with parameters $n$ and $p \in (0, 1)$. If $E_p[g(T)] = (1 - p)^n \sum_{t=0}^{n} g(t)\binom{n}{t} r^t = 0$ with $r = p/(1 - p)$ for all $p$, the polynomial in $r$ vanishes on $(0, \infty)$, so each $g(t) = 0$; thus $T$ is complete. Non-example: let $X_1, X_2$ be i.i.d. $\mathcal{N}(\theta, 1)$ and $T = (X_1, X_2)$. The function $g(T) = X_1 - X_2$ has $E_\theta[g(T)] = 0$ for all $\theta$, but $g(T) \ne 0$ with probability $1$; so $T$ is not complete. Completeness is a property of the family of laws $\set{P_\theta \circ T^{-1} : \theta \in \Theta}$; enlarging $\Theta$ can only help it.