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:


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:


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)?

  • $\begingroup$ it is easy to build artificial examples where part of the sample is ancilary $\endgroup$
    – Xi'an
    Nov 11, 2018 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, 2018 at 16:39

1 Answer 1


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 $$\xi_0=\sum_{i=1}^n \mathbb{I}_{y_i=0}\qquad \xi_1=\{y_i; y_i\ne 0\}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.