# Minimal sufficient statistic whose dimension is less than dimension of parameter

Consider following example:

Suppose $X\sim N(0, \sigma^2)$, consider a random sample of size one from this population. Clearly $X$ is sufficient statistic but $|X|$ is minimal sufficient statistic, which is also complete.

This example shows that if a sufficient statistic has same dimension as complete sufficient statistic then sufficient statistic may not be complete.

I wonder if there exist an example where minimal sufficient statistic has dimension is less than dimension of parameter.

An example where the minimal sufficient statistic has dimension less than the dimension of parameter: a single observation from $\rm{Beta}(\alpha,\beta)$-distribution.