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

Consistency with a fixed nuisance term

Let {} be i.i.d. normal with mean and variance 1, independent of a standard Cauchy random variable W. Which of the following statistics are consistent for ?

  1. A.n⁻ ✓
  2. B.n⁻
  3. C.n⁻ ₋✓
  4. D.n⁻ ✓

Solution

(1), (3) are sample means of n i.i.d. terms. (2) converges to in probability since W is a fixed random variable, so consistency is preserved.

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

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

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