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In Casella Berger's Statistical Inference, they observe that it is rare to find 'a sufficient statistic with dimension smaller than the sample' (section 6.2.1). Although rare, are there examples of non-exponential distributions with dimension reducing sufficient statistics available? The passage in question effectively rules out the order statistics as being a sufficient statistic that reduces dimension in the manner discussed.

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    $\begingroup$ Uniform distribution on $(0,\theta)$ is an example. $\endgroup$ – kjetil b halvorsen Jun 15 '16 at 13:16
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Yes, uniform distribution on $(0,\theta)$ ($\theta>0$) is an example. In this case we can write the density function as $$ f(x; \theta) = \frac1{\theta}\cdot I(0 < x < \theta) $$ where $I$ is the indicator function. Then the likelihood function from an iid sample can be written $$ L(\theta) = \prod_{i=1}^n f(x_i;\theta) = \prod_i \frac1{\theta} I(0 < x_i < \theta) = \\ \frac1{\theta^n} \prod_i I(0 < x_i < \theta) = \\ \frac1{\theta^n} I(0 < \max_i x_i < \theta) $$ and now you can invoke the factorization theorem and conclude that $ \max_i x_i $ is a sufficient statistic.

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