required. 

When CLDR holds for q = 2 , if α = 1 at . 05 level we get max p = 11 , for q = 3 and α = 1 we get max p = 17 . 

These computations show that in many situations the standard test is conservative. 

## 6 An adaptive strategy for determining the asymptotic distribution and the bounds 

From (26) it follows that the asymptotic distribution requires the knowledge of ¯ G ¯ θ and Λ ¯ $_{θ}$. For bounds determining the lowest value on the diagonal of Λ ¯ θ is sufficient. The construction in proof of Lemma 4.3 makes it clear that the main issue for finding the elements of Λ ¯ θ is deciding on the lowest order of the homogeneous polynomial that enters non-trivially into a polynomial (that represents a matrix entry or a determinant of a polynomial matrix), to find ¯ G ¯ θ homogeneous polynomials (their coefficients) of the corresponding lowest orders would have to be consistently estimated. 

## 6.1 Adaptive estimation of polynomial functions and orders 

For ¯ θ consider P ¯ $_{θ}$( θ ) that is a polynomial function of order m$_{P}$ in components of a p × 1 vector ¯ θ with the representation in terms of components of θ − ¯ θ 

