test. Denote by α the smallest α$_{i}$ (usually α$_{1}$ ) in Definition 4.1. Below we show that 1 (1+ α ) 2 ‖ Z ‖ 2 (distributed 1 (1+ α ) 2 χ 2 p under normality) provides a uniform upper bound on the asymptotic null-distribution always under conditions of the Theorem 4.1(a), i.e. when CLDR property holds. If α = 0 , then there may be no singularity in which case with normality the usual asymptotic χ 2 q distribution holds; in general the overall uniform bound with or without singularity under Theorem 4.1.( a ) is χ 2 $_{p}$. It is possible to improve on the χ 2 p bound when α ≥ 1; sometimes the form of the restrictions may provide α ≥ 1. When this is not the case, it may be possible to establish that α ≥ 1 by the adaptive strategy proposed in the next section that would eliminate the possibility that α = 0 . However, for testing it may be sufficient to bound the distribution in the tail rather than everywhere, and so uniform bounds on critical values are also of interest. Gouri´ eroux and Jasiak (2013) discuss a bound on critical values for a test of a determinant. Here in Theorem 5.1 we first establish general bounds on an asymptotic distribution derived for a particular vector of true parameter values, when there may be a singularity at that value. We separately examine the case of one restriction. We also examine a relation between critical values at different α. In the case of one restriction it is possible to provide the number of vari- ables for which generally the standard critical values deliver a conservative test. 29