What is exponential family criterion to test the sufficiency and completeness of an estimator?

Question

I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families:

Let $X_{1},X_{2},...,X_{n}$ be iid observations from an exponential family with PDF or PMF of the form$$f(x|\theta)=h(x)c(\theta)exp\left ( \sum_{j=1}^{k} w(\theta_j)t_j(x) \right )$$where $\theta=(\theta_1,...,\theta_k)$, then $$T(X)=\left ( \sum_{i=1}^{n}t_1(X_i),\sum_{i=1}^{n}t_2(X_i),....,\sum_{i=1}^{n}t_k(X_i) \right )$$is complete as long as the parameter space $\Theta$ contains an open set in $R^k$.

Can someone explain the above definition in simple language? I can apply it in exponential family distributions like gamma, Binomial &tc. But I want to understand what is the significance of this line as long as the parameter space $\Theta$ contains an open set in $R^k$.

Answer 1

The significance of that line is that if you can verify that the parameter space contains an open set in $R^k$, you know, without any further work, that the sufficient statistics $T(X)$ are also complete. That is usually a lot easier to do than trying to apply the definition of completeness directly.

Completeness is a nice property, but not of overwhelming importance. One consequence is that if you have a complete sufficient statistic, you can construct a UMVUE estimator based upon it (Lehmann-Scheffe). See also: What are complete sufficient statistics?.