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Part CCSIR NET June 2024two-lower-bounds-mean-take-the-max

Two lower bounds mean take the max

Let X1,…,Xn(n≥2)X_{1}, \dots, X_{n} (n \ge 2) be a random sample from aU(−θ,2θ)a U(-\theta, 2\theta) distribution, where θ>0\theta > 0 is an unknown parameter. Let Xˉ=(1/n)∑i{\bar{X}} = (1/n)\sum_{i}₌1nXi,X(1)=min⁡_{1}^{n}X_{i}, X_{(1)} = \min{X1,…,XnX_{1}, \dots, X_{n}} and X(n)=max⁡X_{(n)} = \max{X1,…,XnX_{1}, \dots, X_{n}}. Which of the following statements are true?

  1. A.Maximum likelihood estimator of θ\theta is min{X(1),X(n)/2X_{(1)}, X_{(n)}/2}
  2. B.Maximum likelihood estimator of θ\theta is max{−X(1),X(n)/2-X_{(1)}, X_{(n)}/2}✓ Correct
  3. C.Method of moments estimator of θ\theta is 2X̄✓ Correct
  4. D.Method of moments estimator of θ\theta is 2X̄/3

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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

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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