Question

Let X₁ , X₂ , … , X_n(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

The Correct Option is B

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