Part CCSIR NET June 2025location-and-scale-are-the-same-theta-so-the-variance-is-2-theta-squared

Location and scale are the same theta so the variance is 2 theta squared

Let $X_{1}, X_{2}, \dots, X_{n} (n \geq 2)$ be a random sample from a continuous distribution with the probability density function f(x| $θ) =$ $(1/2 θ) e^{- ∣ x - θ ∣/ θ}$ , $x \in R$ , where $θ (> 0)$ is an unknown parameter. Let $U_{n} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}, V_{n} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}^{2}$ , and $S_{n}^{2} = (1/ (n - 1)) \sum_{i}$ ₌ $_{1}^{n} (X_{i} - U_{n})^{2}$ . Then, which of the following statements are true?

A. $U_{n}$ is an unbiased estimator of $θ$ ✓
B. $S_{n}^{2}$ is an unbiased estimator of $θ^{2}$
C. $(1/3) V_{n}$ is a consistent estimator of $θ^{2}$ ✓
D.The statistic $(U_{n}, V_{n})$ is complete

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: A sufficient statistic is complete

Location and scale are the same theta so the variance is 2 theta squared

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