form of this statement under weaker assumptions.

\[rateofp\] Although the asymptotic approximations in Theorem \[t1\] require that $p$ is of the same order as $n$, we point out that the non-central $F$-distribution should still give a reasonable approximation to the distribution of the $F$-statistic, i.e., the expression in should be small, even if $p/n$ is very small, and, in particular, if $p$ is fixed while $n$ increases. This situation is further discussed in @Ste15 [p. 31, Section 3.2.2] in a setting where $n\to\infty$, $p$ is fixed and $p/\log d\to 0$. Clearly, the same is not true for the expression in , because the normal approximation to the $F$ is valid only if both degrees of freedom, i.e., $p$ and $n-p-1$, are large. The statement regarding in Theorem \[t1\] coincides with the conclusion of Theorem 1 in @Zho11a obtained for the correctly specified Gaussian error case. Moreover, the Gaussian approximation in has the advantage that it is easier to interpret than the more complicated distribution function of the non-central $F$-distribution in ; see also the discussion in @Ste16a [Remark 2.4].

Simulation analysis {#numericalresults}
===================

Theorem \[t1\] is an asymptotic result. In this section, we study a range of non-asymptotic scenarios through 