since for similar matrices the eigenvalues are the same, so eigenvalues of ( ¯ G ( Z ) ¯ G ( Z ) ′ ) 1 2 Λ ( ¯ G ( Z ) ¯ G ( Z ) ′ ) − 1 2 are the same as for Λ regardless of Z and the norm is given by the largest eigenvalue, and finally, where for every value of Z the corresponding constant matrix ¯ G $^{′}$( Z ) ( ¯ G ( Z ) ¯ G $^{′}$( Z ) ) − 1 ¯ G ( Z ) is a projection and thus its norm is always bounded by 1. Proof of Lemma 5.1. Express the p.d.f. of χ 2 $_{p$_{1}$}$/α$_{1}$ : and similarly for χ 2 $_{p$_{2}$}$/α$_{2}$ . The ratio p.d.f.$_{χ}$$_{2}$ p$_{1}$ $_{/α$_{1}$}$( y ) p.d.f.$_{χ}$$_{2}$ p$_{2}$ $_{/α$_{2}$}$( y ) is Since α$_{2}$ > α$_{1}$ for large enough y this expression is larger than 1. Proof of Lemma 6.1. First consider ˆ P ( i$_{1}$, ..., i$_{p}$, ¯ θ ) as defined in (30) . By polynomial structure and the convergence rate in Assumption 2.3. ˆ P ( i$_{1}$, ..., i$_{p}$, ¯ θ ) = P ( i$_{1}$, ..., i$_{p}$, ¯ θ ) + O$_{p}$ ( λ − $^{1}$) . Two consequence are (a) from Assumption 2.4 then ˆ P ( i$_{1}$, ..., i$_{p}$, ¯ θ, V ) − ¯ P ( i$_{1}$, ..., i$_{p}$, ¯ θ, V ) →$_{p}$ 0; (b) when P ( i$_{1}$, ..., i$_{p}$, ¯ θ ) = 0 , Pr ( ˆ P i$_{1}$, ..., i$_{p}$, ¯ θ ) = 0 ) → 1 by construction (30) . Since ˆ k$_{P}$ − 1 can be de- fined as the highest order of polynomial with ˆ P ( i$_{1}$, ..., i$_{p}$, ¯ θ ) = 0 it follows 44 2 p$_{2}$ − p$_{1}$ 2 ( Γ ( p$_{2}$ 2 ) / Γ ( p$_{1}$ 2 )) α$^{p$_{2}$}$ 2 − 1 α$^{p$_{1}$}$ 2 − 1 1 y p$_{1}$ − p$_{2}$ 2 exp ( y 2 ( α$_{2}$ − α$_{1}$ ) ) . ∥ ( ¯ G ( Z ) ¯ G ( Z ) $^{′}$) − 1 2 ¯ G ( Z ) Z ∥ ∥$^{2}$ = ( Z ′ ¯ G ( Z ) ′ ( ¯ G ( Z ) ¯ G ( Z ) $^{′}$) − 1 ¯ G ( Z ) Z ) p.d.f.$_{χ}$ 2 $_{$_{p$_{1}$}$/α$_{1}$}$( y ) = α$_{1}$ 2 p$_{1}$/ $^{2}$Γ ( p$_{1}$ 2 ) exp( − α$_{1}$y/ 2) ( α$_{1}$y ) p$_{1}$ 2 − 1 ,