Question

Let 𝑋1,𝑋2, … , 𝑋𝑛 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

Answer: (B)

$2\sqrt3\^σ+\^θ=\overline{X}-4\sqrt3S$

Answer 2

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