Discussion
Y1, Y2, Y3 are three random variables with E(Y1) = Q1 − Q2 + Q3, E(Y2) = Q1 and E(Y3) = Q3 − Q2. Then which of the following statements is not true ?
Answer: B
The statement not true is all three parameters Q1, Q2, Q3 are estimable.
The expectations are \(Q_{1} - Q_{2} + Q_{3}\), \(Q_{1}\) and \(Q_{3} - Q_{2}\). The first is the sum of the other two, so only two independent linear functions are estimable, spanned by \(Q_{1}\) and \(Q_{3} - Q_{2}\). With rank 2 and three parameters, they cannot all be estimable.
The other statements are true. \(Q_{2} - Q_{3} = -E(Y_{3})\) is estimable. \(Q_{1}\) is estimable, but \(Q_{2}\) and \(Q_{3}\) separately are not, since they appear only as \(Q_{3} - Q_{2}\). And \(2Q_{1} - Q_{2} + Q_{3} = E(Y_{1}) + E(Y_{2})\) is estimable.
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