Question:
If S(x) is any sufficient statistic and M(x) is a minimal sufficient statistic for the parametric function \(\tau(\theta)\), then which of the following is not correct for M(x) ?
If S(x) is any sufficient statistic and M(x) is a minimal sufficient statistic for the parametric function \(\tau(\theta)\), then which of the following is not correct for M(x) ?
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Minimal sufficiency is about maximum data reduction, minimum variance additionally needs completeness.
Updated On: Jul 4, 2026
- M(x) takes the minimum value among all the sufficient statistics S(x)
- M(x) takes the minimum variance among all the sufficient statistics S(x)
- M(x) is capable of eliminating irrelevant information to maximum extent among all S(x)
- M(x) gives maximum information about \(\tau(\theta)\) among all the sufficient statistics S(x)
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The Correct Option is B
Solution and Explanation
Step 1: A minimal sufficient statistic M(x) is, by definition, a function of every other sufficient statistic S(x), so it induces the coarsest possible partition of the sample space among all sufficient partitions.
Step 2: Because M(x) is obtained by reducing any S(x) as far as possible while still keeping sufficiency, it eliminates redundant or irrelevant information to the greatest extent, this matches option (C).
Step 3: Since sufficiency itself means no information about \(\tau(\theta)\) is lost, every sufficient statistic, including M(x), retains the complete information the sample carries about \(\tau(\theta)\), so option (D) and option (A), which describe M(x) as the smallest such statistic carrying the full information, are consistent with the definition of minimal sufficiency.
Step 4: Minimum variance of an estimator, however, is not guaranteed by minimal sufficiency alone. Reaching a minimum variance unbiased estimator additionally requires the statistic to be complete, as used in the Lehmann-Scheffe theorem. A minimal sufficient statistic need not be complete, so it need not automatically produce minimum variance.
Step 5: Therefore option (B), which claims minimal sufficiency by itself gives minimum variance, is the statement that is not correct.
Final Answer: (B).
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