Start with the representation of the asymptotic distribution from (26): This distribution depends on the singularity properties that are exhibited at the true value ¯ θ, V. As the theorem below states a bound that depends only on α is possible in all cases when CLDR holds. Theorem 5.1. Under the conditions of Theorem 4.1(a), the asymptotic distribution of the Wald statistic under the null (that depends on the singular- ity properties at ¯ θ ) is bounded from above by the distribution of 1 (1+ α ) 2 ‖ Z ‖ $^{2}$; under the normality Assumption 2.3 this bound is 1 (1+ α ) 2 χ 2 $_{p}$. Thus under conditions of the Theorem 5.1 there is always a general upper bound on the distribution of the Wald statistic under the null given by χ 2 $_{p}$. Remark 5.1. When Λ = I implying that ¯ G ( Z ) does not depend on Z and is a q × p rank q matrix of constants the projection is onto a q − dimensional subspace, the limit distribution is standard and is given under normality by χ 2 $_{q}$. In the case of one restriction under normality the upper bound is either the usual χ 2 $_{1}$, if α = 0 , or else for some α > 0 the bound is 1 (1+ α ) $_{2}$χ 2 $_{p}$. If all that is known is that α > 0 , then the bound 1 $_{4}$χ 2 p applies for any such α. 30 5.1 A general uniform upper bound W ( Z ) = Z ′ ¯ G ( Z ) $^{′}$Λ [ ¯ G ( Z ) ¯ G ( Z ) $^{′}$] − $^{1}$Λ ¯ G ( Z ) Z.