Question

Suppose that $X_{1}, X_{2}, \dots, X_{n}$ are independent random variables each drawn from a population having density function $f_{X}(x)=\begin{cases} \frac{1}{\theta} e^{-(x-\mu)/\theta} & \text{if } x \ge \mu \\ 0 & x < \mu \end{cases}$ where $\theta > 0$ and $\mu \in R^{+}$, then maximum likelihood estimate of $(\theta, \mu)$, when both $\theta, \mu$ are unknown is

• A. $(\overline{X}-X_{(1)}, X_{(1)})$
• B. $(\frac{1}{\overline{X}}-X_{(1)}, X_{(1)})$
• C. $(\overline{X}, X_{(1)})$
• D. $(\frac{1}{\overline{X}}, X_{(1)})$

Hint:

For distributions where the support depends on a parameter (like the lower bound $\mu$ here), the MLE of that bound is usually the extreme order statistic ($X_{(1)}$ or $X_{(n)}$).

Answer 1

The Correct Option is A