Part CCSIR NET June 2024umvue-need-not-attain-the-bound

Umvue need not attain the bound

Let $X_{1}, \dots, X_{n} (n \geq 3)$ be a random sample from a distribution having probability density function f(x | $θ) = θ e^{(} - θ x)$ if x > 0, and 0 otherwise, where $θ > 0$ is an unknown parameter. Let $T_{n} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}$ . Which of the following statements are true?

A.Uniformly minimum variance unbiased estimator of $θ$ is (n − 1)/(nT $_{n})$ ✓
B.Cramer-Rao lower bound for the variance of any unbiased estimator of $θ$ is $θ^{2} / n$ ✓
C.Uniformly minimum variance unbiased estimator of $θ$ attains the Cramer-Rao lower bound
D. $(1 - e^{(} - 1/ T_{n}))$ is a consistent estimator of $P_{θ} (X_{1} \leq 1)$ ✓

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

Umvue need not attain the bound

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