# Minimal sufficient statistics of increasing dimensionality (not equal to the number of observations)

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)?

• it is easy to build artificial examples where part of the sample is ancilary Commented Nov 11, 2018 at 2:58
• 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. Commented Nov 20, 2018 at 16:39

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 $$\xi_0=\sum_{i=1}^n \mathbb{I}_{y_i=0}\qquad \xi_1=\{y_i; y_i\ne 0\}$$
Let $$C$$ be the collection of compact convex subsets of $$\mathbb{R}^n$$, and given $$c\in C$$ let $$P_c$$ be the uniform distribution on $$c$$. Then, let $$X_1, X_2, \dotsc, X_n$$ be an iid sample from $$P_c$$. This arises for instance in connection with estimating the volume of a convex set. Now, the natural way to estimate $$c$$ from the sample is to take the convex hull https://en.wikipedia.org/wiki/Convex_hull Given the convex hull, data points in the interior of the convex hull do not give any new information. The convex hull is given by the extreme points. That is, the set of extreme points is a minimal sufficient statistic for the model $$\{ P_c \colon c\in C\}$$.
The dimension of this minimal sufficient statistic is of the order of $$\log(n)$$, so growth is sublinear.