Question

Let X1 , X2 , … , Xn(n ≥ 2) be a random sample from Exp($\frac{1}{\theta}$) distribution, where θ > 0 is unknown. If $\overline{X}=\frac{1}{n}\sum^n_{i=1}X_i$, then which one of the following statements is NOT true ?

• A. $\overline{X}$ is the uniformly minimum variance unbiased estimator of θ
• B. $\overline{X}^2$ is the uniformly minimum variance unbiased estimator of θ2
• C. $\frac{n}{n+1}\overline{X}^2$ is the uniformly minimum variance unbiased estimator of θ2
• D. $Var(E(X_n|\overline{X}))\le Var(X_n)$

Answer 1

Answer: (B)

$\overline{X}^2$ is the uniformly minimum variance unbiased estimator of θ2

Answer 2

The correct option is (B) : $\overline{X}^2$ is the uniformly minimum variance unbiased estimator of θ2.