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

The Correct Option is C

Let's evaluate the provided statements to find the false one in the context of the statistical problem.

We are given that \(X_1, X_2, X_3, X_4\) form a random sample from a normal distribution \(N(\theta, 1)\), where \(\theta\) is an unknown parameter. The sample mean is \( \bar{X} = \frac{1}{4} \sum_{i=1}^{4} X_i\). We also have the function \( g(\theta) = \theta^2 + 2\theta \) and we need to discuss aspects related to the Cramer-Rao Lower Bound (CRLB) for unbiased estimators of \( g(\theta) \).

1. Statement 1: \(L(\theta) = (1 + \theta)^2\) 
2. Statement 2: \(\bar{X} + e^{\bar{X}}\) is a sufficient statistic for \(\theta\)
3. Statement 3: \((1 + \bar{X})^2\) is the uniformly minimum variance unbiased estimator (UMVUE) of \(g(\theta)\)
4. Statement 4: \(\text{Var}((1 + \bar{X})^2) \geq \frac{(1+\theta)^2}{2}\)

Based on the analysis, the statement "(1 + \bar{X})^2 is the uniformly minimum variance unbiased estimator of \(g(\theta)\)" is found false given the expression doesn't immediately satisfy all unbiased estimator criteria for \(g(\theta)\) without additional proofs. CRLB checks suggest required variance properties aren’t confirmed for this transformed mean expression.