Question

Let $X_{1}, X_{2}, \dots, X_{n}$ constitute a random sample of size "n" from a population having density $f_{X}(x)=\begin{cases} e^{-(x-\theta)} & x > \theta \\ 0 & \text{otherwise} \end{cases}$ then
A. $X_{(1)}$ is sufficient for $\theta$

B. $X_{(1)}$ is consistent for $\theta$

C. $X_{(1)}$ is unbiased for $\theta$

D. $MSE(X_{(1)}) = \frac{2}{n^2}$

• A. A, C only
• B. B, C only
• C. A, B, D only
• D. A, B, C, D only

Hint:

$X_{(1)}$ is the MLE for $\theta$ in a shifted exponential. Like many MLEs, it is biased but consistent. To make it unbiased, one would use $X_{(1)} - 1/n$.

Answer 1

The Correct Option is C