content_image/306549.jpg content_image/306548.jpg content_image/306546.jpg content_image/306547.jpg
For one restriction CLDR always holds and ˆ a = ˆ k as defined in Lemma 6.2 provides α with probability approaching 1. 

For several restrictions estimate ˆ k$_{det}$, ˆ G ¯ $_{θ}$( y ) as defined in Theorem 6.1. and then if CLDR property holds it is sufficient to define the estimate of α as the smallest diagonal element of ˆ Λ ¯ θ as in Theorem 6.1. This estimator will equal the true α with probability approaching 1. 

Use the bound 1 (ˆ a +1) 2 χ 2 $_{p}$. 

## 7 Appendix: Proofs 

Proof of lemma 4.1. We first note that for the submatrix ˜ G$_{l}$ ( y ) defined by (13), 

is a submatrix of ¯ G ( y ) . Then 

exists and since S is nonsingular 

Then this is 

<img src='content_image/306549.jpg'>

<img src='content_image/306548.jpg'>

<img src='content_image/306546.jpg'>

<img src='content_image/306547.jpg'>

