Discussion
Suppose \(\{X_{1}, X_{2}, \ldots, X_{n}\}\) is a random sample of size \(n\) from uniform \(U(\theta - 1, \theta + 1)\) distribution. Then which of the following is not true ?
Answer: A
The statement that is not true is that \(X_{(n)}\) is consistent for \(\theta\).
The upper end of the support is \(\theta + 1\), and the sample maximum converges to it: \(X_{(n)} \to \theta + 1\) in probability. So \(X_{(n)}\) is consistent for \(\theta + 1\), not for \(\theta\), and \(X_{(n)} - 1\) is consistent for \(\theta\) (D).
The distribution is symmetric about \(\theta\), so the sample mean (by the law of large numbers) and the sample median (which converges to the population median \(\theta\)) are both consistent, making B and C true.
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