Part CCSIR NET December 2025both-support-constraints-are-lower-bounds-on-theta-so-the-mle-is-their-max-not-their-min

Both support constraints are lower bounds on theta so the mle is their max not their min

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from Uniform $[-3\theta/2, \theta/2]$ distribution, where $\theta>0$ is an unknown parameter. Let ${\bar{X}}, X_{(1)}$ and $X_{(n)}$ denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?

A.The method of moments estimator of $\theta$ is −2X̄✓
B.The method of moments estimator of $\theta$ is 2X̄
C.The maximum likelihood estimator of $\theta$ is min{ $-2X_{(1)}/3, 2X_{(n)}$ }
D.The maximum likelihood estimator of $\theta$ is max{ $-2X_{(1)}/3, 2X_{(n)}$ }✓

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: The MLE is unbiased

Both support constraints are lower bounds on theta so the mle is their max not their min

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