Discussion
Let \(X_{1}, X_{2}, \ldots, X_{2n+1}\) be a random sample from a uniform distribution on the interval (\(\theta - 1\), \(\theta + 1\)). Let
\(T_{1} = \overline{X}\) (sample mean on 2n + 1 observations)
\(T_{2} = \tilde{X}\) (sample median on 2n + 1 observations)
\(T_{3} = \dfrac{T_{1} + T_{2}}{2}\)
be three estimators of \(\theta\). Then which of the following statements is correct ?
Answer: B
The correct statement is that \(T_{1}\) is consistent for \(\theta\).
\(T_{1} = \overline{X}\) has mean \(\theta\) and variance \(\frac{1}{3(2n + 1)} \to 0\) (the uniform on an interval of length 2 has variance \(\frac{1}{3}\)), so it is consistent.
The others fail. The sample median is also consistent for the centre \(\theta\) of a symmetric distribution, so option A is wrong. By symmetry about \(\theta\), the mean, the median and their average are all unbiased, so 'only \(T_{1}\)' is wrong. The sufficient statistic for this family is the pair (smallest, largest observation), and the median is not sufficient.
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