content_image/300928.jpg content_image/300934.jpg
In the denominator we have 

<img src='content_image/300928.jpg'>

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 

<img src='content_image/300934.jpg'>

Thus CLDR property plays a very important role in the existence of a limit for the Wald statistic. 

## 4.3 The limit distribution of 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. 

