Section 4d

4.5 Canonical Form for H

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Problem 4d.1

Express the hypotheses in Examples 4.4 and 4.5 in canonical form.

Problem 4d.2

Suppose that we have $n_{1}$ observations on $w_{1}, w_{2}, \ldots , w_{p-1}$ and $U$ , giving the model

U_{i} = \gamma _{0}^{(1)}+\gamma _{1}^{(1)} w_{i 1}+\cdots +\gamma _{p-1}^{(1)} w_{i, p-1}+\eta _{i} \quad \left(i = 1,2, \ldots , n_{1}\right)

We are now given $n_{2}( > p$ ) additional observations which can be expressed in the same way, namely,

\begin{aligned} U_{i} = & \gamma _{0}^{(2)}+\gamma _{1}^{(2)} w_{i 1}+\cdots +\gamma _{p-1}^{(2)} w_{i, p-1}+\eta _{i} \\ & \left(i = n_{1}+1, n_{2}+2, \ldots , n_{1}+n_{2}\right) \end{aligned}

Derive an $F$ -statistic for testing the hypothesis $H$ that the additional observations come from the same model.