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Restricting the attention to the case of fixed parameters support, it's my understanding that (minimal) sufficient statistics of fixed dimensionality, i.e. a fixed number of of them, exists in, and only in, the case of exponential families.

Looking outside, I found the case of the Cauchy distribution:

https://math.stackexchange.com/questions/2975830/minimal-sufficient-statistics-for-cauchy-distribution

where "reduction" is a achieved by the sorting, which is not invertible. I don't find this example too interesting as order statistics seems to be always sufficient for iid data and the dimensionality is equal to the dataset size.

I also found the following (very) related question:

https://math.stackexchange.com/questions/149065/examples-of-sufficient-statistics-for-non-exponential-family-distributions

but I fail to understand how the first example doesn't have fixed dimensionality of 1, while the second example is too loosely defined.

What is a known example of a distribution whose minimal sufficient statistics dimensionality isn't fixed nor equal to the dataset size (ideally sublinear)?

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  • $\begingroup$ it is easy to build artificial examples where part of the sample is ancilary $\endgroup$ – Xi'an Nov 11 '18 at 2:58
  • $\begingroup$ In the provided link on math.stackexchange, the answer is not correct in that the dimension of the sufficient statistic remains equal to one. Outside exponential and quasi-exponential families, I know of no example that does not grow linearly with the dataset. $\endgroup$ – Xi'an Nov 20 '18 at 16:39
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An example inspired from the link in the question is the observation of a sample $$\mathbf{y}=(\zeta_1 x_1,\ldots,\zeta_n x_n)\qquad \zeta_i\sim\mathcal{B}(p)\quad x_i\sim \mathcal{T}_3(\mu,\tau)$$ since a sufficient statistic is made of $$\xi_0=\sum_{i=1}^n \mathbb{I}_{y_i=0}\qquad \xi_1=\{y_i; y_i\ne 0\}$$

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