Theory and Methods of Statistics

Hence by an application of Lemma B.7.2 in Appendix B, we can conclude that, conditionally on $\tilde{Y}$, $SS{E}_F/{\sigma }^2\sim {\chi }_{\left(a-1\right)\left(b-1\right)-1}^2\left({\delta }^2-{\delta }_1^2\right)$ and SSE_F/σ² is independent of $\left[SS{E}_R-SS{E}_F\right]/{\sigma }^2$. This allows us to construct a test for H₀: θ = 0 vs H₁: θ≠0 using the F-statistic

$\begin{array}{ll}\hfill F& =\frac{SS{E}_R-SS{E}_F}{MS{E}_F}=\frac{{\widehat{\theta }}^2{S}_{\alpha }{S}_{\beta }}{MS{E}_F},\text{where}\hfill \\ \hfill MS{E}_F& =\left[\sum _i\sum _j{\tilde{\epsilon }}_{ij}^2-{\widehat{\theta }}^2{S}_{\alpha }{S}_{\beta }\right]/\left[\left(a-1\right)\left(b-1\right)-1\right],\hfill \end{array}$

and $F\sim F_{1,\left(a-1\right)\left(b-1\right)-1}$ under H₀.

Example ...