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Part CCSIR NET December 2024the-expression-simplifies-to-twice-the-usual

The expression simplifies to twice the usual

Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample from N(θ,1)N(\theta, 1) distribution, where θ∈R\theta \in \mathbb{R} is unknown. Let δn\delta_{n} be the Bayes estimator of θ\theta, under the squared error loss function L(θ,a)=(a−θ)2,a,θ∈RL(\theta, a) = (a - \theta)^{2}, a, \theta \in \mathbb{R} and the prior distribution N(1, 2). If (1/n)[(2n+1)δn−1−2nθ](1/\sqrt{n})[(2n + 1)\delta_{n} - 1 - 2n\theta] converges in distribution to a random variable Z, as n→∞n \to \infty, then which of the following statements are true?

  1. A.δn\delta_{n} converges in probability to θ\theta, as n→∞n \to \infty, for all θ∈R\theta \in \mathbb{R}✓ Correct
  2. B.Z follows normal distribution.✓ Correct
  3. C.E(Z4)=48E(Z^{4}) = 48✓ Correct
  4. D.E(Z2)=1E(Z^{2}) = 1

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: The MLE is unbiased

More on this topic

The chapter behind this: Maximum likelihood and the method of moments — free to read

From Estimation › MLE and method of moments

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