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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.

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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.

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    $\begingroup$ Is the minimal sufficient statistic x itself? $\endgroup$
    – Tan
    Commented Nov 7, 2020 at 3:15
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    $\begingroup$ @Tan: Yes, it is $\endgroup$
    – kjetil b halvorsen ♦
    Commented Jan 29, 2021 at 16:40

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