📂Mathematical Statistics

Complete Statistics of the Exponential Family of Probability Distributions

Theorem ¹

Given a parameter $\mathbf{\theta} = \left( \theta_{1} , \cdots , \theta_{k} \right)$ and the probability density function or probability mass function of a random sample $X_{1} , \cdots , X_{n}$ follows an exponential family distribution as shown below. $$f(x; \mathbf{\theta}) = h(x) c (\mathbf{\theta}) \exp \left( \sum_{i=1}^{k} w_{i} \left( \theta_{j} \right) t_{i} (x) \right)$$ Then the following statistic $T$ is a complete statistic. $$T \left( \mathbf{X} \right) = \left( \sum_{i=1}^{n} t_{1} \left( X_{i} \right) , \cdots , \sum_{i=1}^{n} t_{k} \left( X_{i} \right) \right)$$

Proof

It is trivial by the uniqueness of the Laplace transform.

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