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GATE 2025 ST – Question 18

Estimation · Sufficiency and Completeness · 1 mark · Multiple choice

T is complete sufficient for a family of distributions, U is sufficient, and T≥0 almost surely under every member. Which is not necessarily true?

  1. T² is complete.
  2. T² is minimal sufficient.
  3. T is a function of U.
  4. U is a function of T.

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Correct answer: (D) U is a function of T.

Explanation

A complete sufficient statistic is minimal sufficient under the standard dominated-family setting, so T is determined by any sufficient U. Squaring is one-to-one on the almost-sure nonnegative range, preserving sufficiency, completeness and minimality. U can contain extra information, such as the full labeled sample, that is not determined by T. Thus **D**.

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