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Part CCSIR NET June 2023consistency-with-a-fixed-nuisance-term

Consistency with a fixed nuisance term

Let {Xi:1≤i≤2nX_{i} : 1 \le i \le 2n} be i.i.d. normal with mean μ\mu and variance 1, independent of a standard Cauchy random variable W. Which of the following statistics are consistent for μ\mu?

  1. A.n⁻1∑i=1n^{1} \sum_{i=1}^{n} XiX_{i}✓ Correct
  2. B.n⁻1∑i=12n^{1} \sum_{i=1}^{2n} XiX_{i}
  3. C.n⁻1∑i=1n^{1} \sum_{i=1}^{n} X2iX_{2i}₋1_{1}✓ Correct
  4. D.n⁻1(∑i=1n^{1}(\sum_{i=1}^{n} Xi+W)X_{i} + W)✓ Correct

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

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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