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✓ Correct
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{ $-2X_{(1)}/3, 2X_{(n)}$ }✓ 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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

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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