Part CCSIR NET December 2024overlapping-index-ranges-are-not-independent

Overlapping index ranges are not independent

Let $X_{1}, X_{2}, \dots, X_{13}$ be independent and identically distributed (i.i.d.) Poisson $(θ)$ random variables, where $θ > 0$ . Then, which of the following statements are true?

A. $(\sum (i = 1..8) X_{i}) (\sum (i = 8..13) X_{i})$ is an unbiased estimator of $48 θ^{2}$ .
B. $(1/13) \sum (i = 1..13) X_{i}$ is method of moments estimator of $θ$ .✓
C.There does not exist any unbiased estimator of $e^{(} - 7 θ)$ .
D. $(e^{(} X_{7}), \sum (i = 1..6) X_{i}, \sum (i = 8..13) X_{i})$ is a sufficient statistic for $θ$ .✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: A sufficient statistic is complete

Overlapping index ranges are not independent

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