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Counterexample

Is this true?

A sufficient statistic is complete

No — it is false.

The counterexample

X(n)(X_{(n)} (or (X(1),X(n)))(X_{(1)}, X_{(n)})) for Uniform(−θ,θ)(-\theta, \theta)

Why it fails

Sufficient but not complete: symmetry gives non-zero functions with zero expectation.

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Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

See real exam options built on this mistakeDrill statements like this
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Go deeper· Explained in full in Sufficiency, completeness and the UMVUE machine· Asked in CSIR NET December 2025· From Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao

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