In the denominator we have det[ S ( λ ) G ( y/λ )Ω G $^{′}$( y/λ ) S $^{′}$( λ )] = λ 2Σ α$_{i}$ det[ SG ( y/λ )Ω G $^{′}$( y/λ ) S $^{′}$] = λ 2[Σ α$_{i}$ − ¯ α $^{]}$λ2¯ $^{α}$det[ SG ( y/λ )Ω G $^{′}$( y/λ ) S $^{′}$] . Thus as λ → ∞ , when the CLDR property is not fulfilled, while λ 2¯ $^{α}$det[ SG ( y/λ )Ω G $^{′}$( y/λ ) S $^{′}$] has a finite limit for every Ω, λ 2[Σ α$_{i}$ − ¯ α ] converges to zero and Thus CLDR property plays a very important role in the existence of a limit for the Wald statistic. Define y = A ( θ − ¯ θ ) for some non-degenerate matrix A ; with this substitu- tion under the assumption g ( ¯ θ ) = 0 the polynomial function g ( θ ) becomes g ( A − $^{1}$y + ¯ θ ) = g ¯ $_{θ}$( y ) , a polynomial function with g ¯ $_{θ}$(0) = 0 . The Jacobian polynomial matrix G gets multiplied by the nonsingular matrix A − 1 to pro- vide the new Jacobian G ¯ θ ( y ) with respect to y and G ( θ ) = G ¯ $_{θ}$( y ) A. Note the role that the nonsingular matrix A plays: it does not change the order of polynomial function g ; if CLDR property holds for G ¯ θ defined with some nonsingular A, it holds for any other nonsingular A. In the notation for the function and the Jacobian we do not emphasize then the role of A. 25 4.3 The limit distribution of the Wald statistic W ( y, g, λ, Ω) → λ →∞ ∞ . (24)