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.

When the sample size is only 1 and the parameter dimensionality is large, what can you do? You put your head down and wait for more data to come.

  • 1
    $\begingroup$ Is the minimal sufficient statistic x itself? $\endgroup$ – Tan Nov 7 '20 at 3:15
  • 1
    $\begingroup$ @Tan: Yes, it is $\endgroup$ – kjetil b halvorsen Jan 29 at 16:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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