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I am struggling to understand the following definition from casella and Berger about the exponential family sufficiency and completeness:

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

$T(X)=\left ( \sum_{i=1}^{n}t_1(X_i),\sum_{i=1}^{n}t_2(X_i),....,\sum_{i=1}^{n}t_k(X_i) \right )$ complete as long as the parameter space $\Theta$ contains an open set in $R^k$.

Can someone explain the above definition in a simple language? I can apply it in all the exponential family distribution like gamma, Binomial Etcetera. But I want to understand the what is the significance of this line as long as the parameter space $\Theta$ contains an open set in $R^k$. It will be very beneficial for me. Thanks in advance.

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  • $\begingroup$ 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/…). $\endgroup$ – Fabian Werner Jan 10 '18 at 10:08
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    $\begingroup$ @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! $\endgroup$ – jbowman Jan 10 '18 at 18:02
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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?.

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