Part CCSIR NET December 2023shifted-exponential

Shifted exponential

Let $X_{1}, \dots, X_{n}$ be a random sample with density f(x| $θ) =$ $e^{θ - x}$ for $x \geq θ$ , where $θ \in R$ 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 $θ$
B. $X_{(1)}$ is the maximum likelihood estimator of $θ$
C. $X_{(1)} - 1/ n$ is the uniformly minimum variance unbiased estimator of $θ$
D. $X_{(1)}$ is a sufficient statistic for $θ$

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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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