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Part CCSIR NET December 2024injective-functions-inherit-sufficiency

Injective functions inherit sufficiency

Let X, Y, Z be independent and identically distributed (i.i.d.) random variables, each following Bernoulli(θ),0<θ<1(\theta), 0 < \theta < 1. Then, which of the following statements are true?

  1. A.(79X+2024Y,23Z2)(79X + 2024Y, 23Z^{2}) is a sufficient statistic for θ\theta.✓ Correct
  2. B.13(X+Y+Z)313(X + Y + Z)^{3} is a minimal sufficient statistic for θ\theta.✓ Correct
  3. C.2(X + Y + Z) is a complete sufficient statistic for θ\theta.✓ Correct
  4. D.(X − 2Y + Z) is an ancillary statistic.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests property not inherited.

See pricing

50 are analysed free — try those first.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

Related counterexample: A sufficient statistic is complete

More on this topic

The chapter behind this: Sufficiency, completeness and the UMVUE machine — free to read

From Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao

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