Are there examples of non exponential family distributions with sufficient statistics?

Question

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.

Answer 1

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.