My textbook gives the following theorem for exponential families:

Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(\theta)t(x))$$ Then the statistic $T(X) = \sum_{i=1}^n t(X_i)$ is complete.

But I'm unsure how to interpret this. It does not mention whether the family is curved, full rank, or of minimal dimension. And I feel like these conditions will certainly affect whether the statistic $T(X)$ is complete or not.

Am I right, and that extra conditions are not explicitly stated? Or is simply being an exponential family enough to imply completeness of $T(X)$?


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