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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 X1,X2,…,Xn(n≥2)X_{1}, X_{2}, \dots, X_{n} (n \ge 2) be a random sample from a continuous distribution with the probability density function f(x|θ)=\theta) = (1/2θ)e−∣x−θ∣/θ(1/2\theta)e^{-|x-\theta|/\theta}, x∈Rx \in \mathbb{R}, where θ(>0)\theta (> 0) is an unknown parameter. Let Un=(1/n)∑iU_{n} = (1/n)\sum_{i}₌1nXi,Vn=(1/n)∑i_{1}^{n} X_{i}, V_{n} = (1/n)\sum_{i}₌1nXi2_{1}^{n} X_{i}^{2}, and Sn2=(1/(n−1))∑iS_{n}^{2} = (1/(n-1))\sum_{i}₌1n(Xi−Un)2_{1}^{n}(X_{i} - U_{n})^{2}. Then, which of the following statements are true?

  1. A.UnU_{n} is an unbiased estimator of θ\theta✓ Correct
  2. B.Sn2S_{n}^{2} is an unbiased estimator of θ2\theta^{2}
  3. C.(1/3)Vn(1/3)V_{n} is a consistent estimator of θ2\theta^{2}✓ Correct
  4. D.The statistic (Un,Vn)(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

More on this topic

The chapter behind this: Sufficiency, completeness and the UMVUE machine — free to read

From Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao

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