Part BCSIR NET December 2024the-parameter-moves-both-endpoints

The parameter moves both endpoints

Let $X_{1}, X_{2}, \dots, X_{n} (n \geq 2)$ be a random sample from Uniform $(θ, 1 - θ)$ distribution, where $- \infty < θ < 1/2$ . The maximum likelihood estimator of $θ$ is

A.min{1 − min{ $X_{1}, X_{2}, \dots, X_{n}$ }, max{ $X_{1}, X_{2}, \dots, X_{n}$ }}
B.max{1 − min{ $X_{1}, X_{2}, \dots, X_{n}$ }, max{ $X_{1}, X_{2}, \dots, X_{n}$ }}
C.min{min{ $X_{1}, X_{2}, \dots, X_{n}$ }, 1 − max{ $X_{1}, X_{2}, \dots, X_{n}$ }}✓
D.max{min{ $X_{1}, X_{2}, \dots, X_{n}$ }, 1 − max{ $X_{1}, X_{2}, \dots, X_{n}$ }}

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

The parameter moves both endpoints

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