Discussion
Under completely randomized design with model :
\(Ey_{ij} = \mu + \alpha_{i} + \in_{ij}\) \(i = 1, ......, p\)
\(j = 1, ......, n_{i}\)
\(\in_{ij} \sim N(0, \sigma^{2})\) and independently distributed, a parametric function \(c\mu + \sum_{i=1}^{p} d_{i}\alpha_{i}\) where \(c\) and \(d_{1}, ......., d_{p}\) are known constants is estimable if and only if :
Answer: D
The function is estimable if and only if \(\sum d_{i} = c\).
The estimable functions are linear combinations of the expected observations \(E(y_{ij}) = \mu + \alpha_{i}\). A combination \(\sum_{i} l_{i}(\mu + \alpha_{i})\) equals \(\left(\sum l_{i}\right)\mu + \sum l_{i}\alpha_{i}\), so it has the form \(c\mu + \sum d_{i}\alpha_{i}\) with \(d_{i} = l_{i}\) and \(c = \sum d_{i}\).
Conversely any function with \(c = \sum d_{i}\) is such a combination. Contrasts (\(c = 0\), \(\sum d_{i} = 0\)) are the special case; \(c = 0\) alone, or \(\sum d_{i} = 0\) with \(c \gt 0\), break the required equality, and \(\mu\) itself (\(c = 1\), all \(d_{i} = 0\)) is not estimable.
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