Question

Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\ (n\geq2)\) from a \(N(0,\sigma^2)\) distribution, where \(\sigma>0\) is an unknown parameter. Which one of the following statements is true?

• A. \(\sum_{i=1}^{n}X_i\) is a sufficient statistic for \(\sigma^2\)
• B. \(\sum_{i=1}^{n}X_i^2\) is a sufficient statistic for \(\sigma^2\)
• C. \(\sum_{i=1}^{n}X_i\) is a complete statistic
• D. \(\frac{1}{n}\sum_{i=1}^{n}X_i^2\) is a biased estimator of \(\sigma^2\)

Hint:

For normal distributions with known mean and unknown variance, the sufficient statistic for variance is based on the sum of squared observations.

Answer 1

The Correct Option is B

Step 1: Look at the joint density.
For $N(0,\sigma^2)$ the joint density is $(2\pi\sigma^2)^{-n/2}\exp\!\left(-\frac{1}{2\sigma^2}\sum x_i^2\right)$.

Step 2: See what carries the data.
The only place the data enter is through $\sum X_i^2$. By the factorization theorem this is sufficient for $\sigma^2$, so (B) is true.

Step 3: Rule out the sum $\sum X_i$.
The density never uses the plain sum, so $\sum X_i$ is not sufficient, and being symmetric in sign it is also not complete. So (A) and (C) fail.

Step 4: Check the variance estimator.
Since $E(X_i^2)=\sigma^2$, the average $\frac1n\sum X_i^2$ has mean $\sigma^2$, so it is unbiased, not biased. (D) fails.

Step 5: Conclude.
The correct statement is option (B).
\[ \boxed{(B)} \]