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

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 $$R^k$$.

• Without knowing anything about this particular problem my guess is that they need the parameter space contain an open set because they want to take derivatives of functions w.r.t. $\Theta$. See Thm. 3.4.2 (at least here: fsalamri.files.wordpress.com/2015/02/…). – Fabian Werner Jan 10 '18 at 10:08
• @FabianWerner - No, it's the "Open Set Condition" and is sufficient for completeness of the sufficient statistic. You like to have completeness because completeness relates to certain properties of the distribution of the sufficient statistics across all $\theta \in \Theta$. It's more related to the ability to integrate rather than to differentiate! – jbowman Jan 10 '18 at 18:02
• This is not a definition rather a sufficient condition (lemma). – Xi'an Nov 27 '19 at 9:59

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.