where g$_{ik}$ ( θ ) represents a homogeneous polynomial of order k , each coefficient A$_{ik}$ ( i$_{1}$, . . . , i$_{p}$ ) is a constant, and m is the maximal order of a polynomial in g ( θ ). Assumption 2.2. We assume that some ¯ θ satisfies a null hypothesis of the form: Assumption 2.3. Assume that { ˆ θ$_{T}$ : T ≥ T$_{0}$ } is a sequence of p × 1 random vectors such that for some positive definite matrix V and a scalar rate sequence λ$_{T}$ → ∞ as T → ∞ convergence in probability holds: where Z is a random p × 1 vector with a known absolutely continuous prob- ability distribution, Q ( ¯ θ ) on R $^{p}$. 1 2 Assumption 2.3a. In addition to Assumption 2.3 λ$_{T}$ = T 2 , Z is a Gaus- sian random vector. ˆ Assumption 2.4. { V$_{T}$ : T ≥ T$_{0}$ } is a sequence of p × p random matrices such that P [ rank ( ˆ V$_{T}$ ) = p ] = 1 , for all T, and λ$_{T}$ V − 1 2 ( ˆ θ$_{T}$ − ¯ θ ) →$_{p}$ Z, (4) plim T →∞ ˆ V$_{T}$ = V (5) where the probability that ˆ V$_{T}$ be positive definite is one for T ≥ T$_{0}$ ( for some T$_{0}$ > 0). H$_{0}$ : g ( θ ) = 0 . (3) 7