Discussion
\(Y_{1}, Y_{2}\) and \(Y_{3}\) are three uncorrelated random variables with common variance \(\sigma^{2}\). Further \(E(Y_{1}) = \theta_{2} - \theta_{1}\), \(E(Y_{2}) = E(Y_{3}) = \theta_{2} + \theta_{3}\). Then :
Answer: B
The true statement is \(\theta_{1} + \theta_{2}\) is not estimable.
A linear function is estimable when it is a linear combination of the expectations \(E(Y_{1}) = \theta_{2} - \theta_{1}\) and \(E(Y_{2}) = E(Y_{3}) = \theta_{2} + \theta_{3}\). For \(a(\theta_{2} - \theta_{1}) + b(\theta_{2} + \theta_{3})\) to equal \(\theta_{1} + \theta_{2}\), the \(\theta_{3}\) coefficient forces \(b = 0\), and then \(-a = 1\), \(a = 1\) contradict each other. So \(\theta_{1} + \theta_{2}\) is not estimable.
The others are estimable, so their 'not estimable' claims are false: \(\theta_{1} - \theta_{2} = -E(Y_{1})\); \(\theta_{2} + \theta_{3} = E(Y_{2})\); and \(\theta_{1} + \theta_{3} = E(Y_{2}) - E(Y_{1})\).
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