In Example 3.3 the limiting distribution is 1 $_{4}$χ 2 $_{p}$; thus this bound can be attained. In the special case p = q and under CLDR ¯ G is invertible a.e. and Z ′ ¯ G ( Z ) $^{′}$Λ [ ¯ G ( Z ) ¯ G ( Z ) $^{′}$] − $^{1}$Λ ¯ G ( Z ) Z = Z ′ ¯ G $^{′}$Λ ¯ G ′− 1 ¯ G − $^{1}$Λ ¯ GZ. Then since the norm of a similar matrix is the same as for Λ . Under normality the bound is 1 (1+ α ) 2 χ 2 $_{q}$. In this case the asymptotic distribution is bounded from above by the usual distribution and under normality the distribution χ 2 q provides a conservative test. 5.2 Bounds on critical values for purely singular cases α ≥ 1 under normality A conservative test for a given level may be given by the standard critical values, even in the non-standard cases considered here since then dominance by the standard distribution is required only in the tail and not everywhere. The following Lemma demonstrates that when the distribution is purely sin- gular ( α ≥ 1) there is always a level, γ$_{0}$, such that using the standard critical values for any γ ≤ γ$_{0}$ provides a conservative asymptotic test. Indeed, there exists γ$_{0}$ such that Pr( 1 (1+ α ) 2 χ 2 p > χ 2 $_{q}$( γ$_{0}$ )) < γ$_{0}$, where χ 2 q ( γ$_{0}$ ) denotes the critical value. The following lemma establishes this tail dominance. 31 W ( Z ) ≤ ‖ Λ ‖ 2 ‖ Z ‖ $^{2}$,