Question

Let 𝑋1,𝑋2, … , 𝑋𝑛 be a random sample from a population having the probability density function
$f(x;μ) =\begin{cases} \frac{1}{2}e-(\frac{x-2μ}{2}), & \quad \text{if }0>2μ,\\\ 0, & \quad Otherwise \end{cases}$
where −∞ < 𝜇 < ∞. For estimating 𝜇, consider estimators
$T_1=\frac{\overline{X}-2}{2}$ and $T_2=\frac{nX_{(1)}-2}{2n}$
where 𝑋̅ =$\frac{1 }{𝑛} ∑^n_{i=1} x_i$ and Xi and X(i)=min{𝑋1, 𝑋2, … , 𝑋𝑛}. Then, which one of the following statements is TRUE?

• A. 𝑇1 is consistent but 𝑇2 is NOT consistent
• B. 𝑇2 is consistent but 𝑇1 is NOT consistent
• C. Both 𝑇1 and 𝑇2 are consistent
• D. Neither 𝑇1 nor 𝑇2 is consistent

Answer 1

Answer: (C)

Both 𝑇1 and 𝑇2 are consistent

Answer 2

The correct option is (C): Both 𝑇1 and 𝑇2 are consistent