Discussion
\(y_{1}, y_{2}, y_{3}, y_{4}\) are four uncorrelated random variables with common variance \(\sigma^{2}\) and \(E(y_{1}) = E(y_{2}) = \theta_{1} + 2\theta_{2} + \theta_{3}\), \(E(y_{3}) = E(y_{4}) = \theta_{1} + \theta_{3}\). Hence :
Answer: B
The correct statement is that \(2\theta_{2} + 3\theta_{3}\) is not estimable.
A linear function is estimable when it is a combination of the expected values. The available expectations are \(\theta_{1} + 2\theta_{2} + \theta_{3}\) and \(\theta_{1} + \theta_{3}\), so every estimable function has the form
\[ (a + b)\theta_{1} + 2a\theta_{2} + (a + b)\theta_{3} \]
with equal coefficients of \(\theta_{1}\) and \(\theta_{3}\). \(2\theta_{2} + 3\theta_{3}\) has coefficients 0 and 3, so it is not estimable.
The others are estimable: \(\theta_{2}\) (\(a = 1/2, b = -1/2\)), \(\theta_{1} + \theta_{3}\) (\(a = 0, b = 1\)) and \(\theta_{1} + \theta_{2} + \theta_{3}\) (\(a = b = 1/2\)). So A, C and D are false.
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