Question

Let 𝑋1, 𝑋2, 𝑋3, 𝑋4 be a random sample of size 4 from an 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ is an unknown parameter. Let 𝑋̅ = $\frac{1 }{4 }∑^4_{i=1} X_i$ , 𝑔(𝜃) = 𝜃 2 + 2𝜃 and 𝐿(𝜃) be the Cramer-Rao lower bound on variance of unbiased estimators of 𝑔(𝜃). Then, which one of the following statements is FALSE?

• A. 𝐿(𝜃) = (1 + 𝜃) 2
• B. 𝑋̅ + 𝑒 𝑋̅ is a sufficient statistic for 𝜃
• C. (1 + 𝑋̅) 2 is the uniformly minimum variance unbiased estimator of 𝑔(𝜃)
• D. 𝑉𝑎𝑟((1 + 𝑋̅) 2 ) ≥ $\frac{(1+θ) ^2}{ 2}$

Answer 1

Answer: (C)

(1 + 𝑋̅) 2 is the uniformly minimum variance unbiased estimator of 𝑔(𝜃)

Answer 2

The correct option is (C): (1 + 𝑋̅) 2 is the uniformly minimum variance unbiased estimator of 𝑔(𝜃)