Chapter 10.3, Problem 29E

When sample sizes are equal (J_i = J). the parameters α₁, α₂,... α₃, of the alternative parameterization are restricted by $\sum {\alpha }_i=0$ . For unequal sample sizes, the most natural restriction is 2i_fa, = 0. Use this to show that

     $E\left(MSTr\right)={\sigma }^2+\frac{1}{I-1}\sum J_i{\alpha }_i{}^2$

What is E(MSTr) when H₀ is true? [This expectation is correct if $\sum J_i{\alpha }_i=0$ is replaced by the restriction $\sum {\alpha }_i=0$  (or any other single linear restriction on the a/s used to reduce the model to I independent parameters), but $\sum J_i{\alpha }_i=0$ simplifies the algebra and yields natural estimates for the model parameters (in particular, ${\alpha }_i={\bar{x}}_i-\bar{x}\mathrm{..})$.]