Question

Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables having $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$. Define \[ W = \frac{1}{2n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} (X_i - X_j)^2. \] Then $W$, as an estimator of $\sigma^2$, is 
 

• A. Biased and consistent
• B. Unbiased and consistent
• C. Biased and inconsistent
• D. Unbiased and inconsistent

Hint:

Consistency depends on large-sample behavior, while bias checks finite-sample deviation from the true parameter.

Answer 1

The Correct Option is A

Step 1: Simplify $W$.
We know that \[ E[(X_i - X_j)^2] = 2\sigma^2. \] So, \[ E(W) = \frac{1}{2n^2} \times n(n-1) \times 2\sigma^2 = \frac{n-1}{n}\sigma^2. \]

Step 2: Analyze bias and consistency.
The estimator is biased because $E(W) \neq \sigma^2$, but as $n \to \infty$, \[ \frac{n-1}{n} \to 1, \] so $W$ becomes consistent.

Step 3: Conclusion.
Thus, $W$ is biased but consistent.