Question

Let 𝑋₁,𝑋₂, … , 𝑋_𝑛 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 X_i and X_(i)=min{𝑋₁, 𝑋₂, … , 𝑋_𝑛}. 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

The Correct Option is C

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