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

Umvue need not attain the bound

Let $X_{1}, \dots, X_{n} (n \ge 3)$ be a random sample from a distribution having probability density function f(x | $\theta) = \theta{}e^(-\theta{}x)$ if x > 0, and 0 otherwise, where $\theta > 0$ 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 $\theta$ is (n − 1)/(nT $_{n})$ ✓ Correct
B.Cramer-Rao lower bound for the variance of any unbiased estimator of $\theta$ is $\theta^{2}/n$ ✓ Correct
C.Uniformly minimum variance unbiased estimator of $\theta$ attains the Cramer-Rao lower bound
D. $(1 - e^(-1/T_{n}))$ is a consistent estimator of $P_\theta(X_{1} \le 1)$ ✓ 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 what the inference means.

See pricing

50 are analysed free — try those first.

The trap it tests

What the inference means

p-values, size, power and error types say specific things. This was not one of them.

Drill statements like this

Related counterexample: A sufficient statistic is complete

Umvue need not attain the bound

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