content_image/326620.jpg content_image/326621.jpg
follows that thus one can determine with probability approaching 1 whether the CLDR property holds and if it does estimate the matrix Λ ¯ θ with prob- ability approaching 1. Then the corresponding consistent estimator, ˆ G ¯ θ ( y ) of ¯ G ¯ θ ( y ) is obtained for A = I. A consistent estimator of the corresponding polynomial, ¯ G ¯ θ ( Ay ) for A = V − 1 2 can be obtained by substituting ˜ y = ˆ V − 1 2 y into the estimator ˆ G ¯ θ ( y ) to obtain ˜ G ¯ $_{θ}$(˜ y ) . 

Theorem 6.1. Under the Assumptions of Theorem 4.1 if (a) the corre- sponding estimated ˆ k$_{det}$ = 0 , then with probability approaching 1 as T → ∞ the distribution of the asymptotic statistic is standard and under normality is χ 2 $_{q}$; if (b) for A = I, ˆ k$_{det}$ ̸ = 0 but the estimated ˆ G ¯ θ ( y ) has deficient rank as T → ∞ , then the statistic diverges to infinity; if (c) with A = I, ˆ k$_{det}$ ≥ 1 and the estimated ¯ G ¯ θ ( y ) satisfies the CLDR property with the estimated matrix ˆ Λ ¯ $_{θ}$, then the limit distribution is consistently estimated by 

<img src='content_image/326620.jpg'>

## 6.3 Conservative tests with adaptively estimated bounds 

From Theorem 5.1 it follows that if the CLDR property holds the bound is provided by 

<img src='content_image/326621.jpg'>

