When is a statistic insufficient according to the factorization theorem?

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Can anyone give me an example of when a statistic is insufficient, using the factorization theorem, if possible?

Thanks.
 
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You need to remember that sufficiency is defined with respect to a parameter and a sample. "The average of the first three observations out of a sample of N>3 observations" is clearly insufficient for calculating the overall mean from the same sample.
 

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