Discussion
Suppose \(\{X_{1}, X_{2}, \ldots, X_{n}\}\) is a random sample of size \(n\) from the \(U(0, \theta)\) distribution. Then :
Answer: C
The consistent estimator is \(2\bar{X}_{n}\).
For \(U(0, \theta)\), \(E(X) = \theta/2\). By the law of large numbers \(\bar{X}_{n} \to \theta/2\) in probability, so \(2\bar{X}_{n} \to \theta\): it is consistent (and unbiased, with variance \(\theta^{2}/(3n) \to 0\)).
The others converge to the wrong value: \(\bar{X}_{n}\) and the sample median both tend to \(\theta/2\), and the minimum \(X_{(1)}\) tends to 0. (The maximum \(X_{(n)}\), not offered, would also be consistent for \(\theta\).)
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