Part CCSIR NET June 2024two-lower-bounds-mean-take-the-max

Two lower bounds mean take the max

Let $X_{1}, \dots, X_{n} (n \geq 2)$ be a random sample from $a U (- θ, 2 θ)$ distribution, where $θ > 0$ is an unknown parameter. Let $\overset{ˉ}{X} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}, X_{(1)} = min$ { $X_{1}, \dots, X_{n}$ } and $X_{(n)} = max$ { $X_{1}, \dots, X_{n}$ }. Which of the following statements are true?

A.Maximum likelihood estimator of $θ$ is min{ $X_{(1)}, X_{(n)} /2$ }
B.Maximum likelihood estimator of $θ$ is max{ $- X_{(1)}, X_{(n)} /2$ }✓
C.Method of moments estimator of $θ$ is 2X̄✓
D.Method of moments estimator of $θ$ is 2X̄/3

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

Two lower bounds mean take the max

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