Part CCSIR NET December 2023shifted-exponential

Shifted exponential

Let $X_{1}, \dots, X_{n}$ be a random sample with density f(x| $\theta) =$ $e^{\theta-x}$ for $x \ge \theta$ , where $\theta \in \mathbb{R}$ is unknown. Define X̄ and $X_{(1)} = \min X_{i}$ . Which of the following statements are true?

A.X̄ is the method of moments estimator of $\theta$
B. $X_{(1)}$ is the maximum likelihood estimator of $\theta$ ✓ Correct
C. $X_{(1)} - 1/n$ is the uniformly minimum variance unbiased estimator of $\theta$ ✓ Correct
D. $X_{(1)}$ is a sufficient statistic for $\theta$ ✓ 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

Shifted exponential

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