Question

Let 𝑋₁,𝑋₂, … , 𝑋_𝑛 be a random sample from a \(u(θ +\frac{σ}{\sqrt3},θ+\sqrt3σ)\) distribution, where 𝜃 ∈ ℝ and 𝜎>0 are unknown parameters. Let 𝑋̅ =\(\frac{ 1}{ 𝑛} ∑^n _{i=1}X_i\) and \(𝑆 =\sqrt \frac{1}{n} ∑^n_{i=1}(X_i-\overline{X})^2\) Let \(\^θ\) and \(\^σ\) be the method of moment estimators of 𝜃 and 𝜎 ,respectively. Then, which one of the following statements is FALSE?

• A. \(\^σ+\sqrt3\^θ=\sqrt3\overline{X}-3s\)
• B. \(2\sqrt3\^σ+\^θ=\overline{X}-4\sqrt3S\)
• C. \(\sqrt3\^σ+\^θ=\overline{X}+\sqrt3\,S\)
• D. \(\^σ-\sqrt3\,\^θ=9\,S-\sqrt3\,\overline{X}\)

Answer 1

The Correct Option is B

The correct option is (B): \(2\sqrt3\^σ+\^θ=\overline{X}-4\sqrt3S\)