interpreted as a signal-to-noise ratio in and depends on $\theta$, $M$, $\Sigma$ and $\sigma^2={{\mathbb E}}[\epsilon^2]$; cf. . If the error $e$ in is Gaussian and independent of $x$, then the $F$-statistic $\hat{F}_n$ is $F$-distributed with parameters $p$, $n-p-1$ and non-centrality parameter $n \Delta$; in that case, we have ${{\mathbb P}}(\hat{F}_n \leq t) =
F_{n,n-p-1,n\Delta}(t)$, where $F_{n,n-p-1,n \Delta}(\cdot)$ denotes the cumulative distribution function (c.d.f.) of the $F$-distribution with indicated parameters. In our present setting, however, the error $e$ in need not be Gaussian and can depend on $x$.

We will show that the distribution of $\hat{F}_n$ can be approximated by an $F$-distribution, uniformly over most parameters in the model. Only for $\epsilon$, $f_{\tilde{z}}$ and $R$, i.e., for the error in and for the density of the standardized explanatory variables as well as the orthogonal matrix in , some restrictions are needed. We will require a moment restriction on $\epsilon/\sigma$, and we will require that $f_{\tilde{z}}$ belongs to one of the classes ${\mathcal F}_{d,k}(D,E)$ introduced earlier. To formulate the restriction on $R$, write ${\mathcal O}_{d}$ for the collection of all orthogonal $d\times d$ matrices and write $\nu_{d}$ for the uniform distribution on that set; i.e., $\nu_{d}$ is the normalized Haar measure on 