OLHA BODNAR 261

tij∼N(0, 1)i=1,...,q,j=1,...,i, andtiiandtijare mutually independently distributed.
Hence, the expectation of the second term is equal to

E(

√

n− 1

√

H ̃( 22 −)b ̃ˆ−^12 (w ̃ˆM;q−wM;q))=

√

H ̃( 22 −)b ̃−^12





E(t 11 )

..

.

E(tqq)



(w ̃

M;q−wM;q)

DenoteA=diag(a 11 ,...,aqq)withaii=

B

(n−p−q−i+ 1 + 1

2 ,

n−p− 1

2

)

B

(n−p−q−i+ 1

2 ,

n−p

2

) ,i=1,q. The first part of

the theorem is proved.

(b) Here we denotec=

√

n− 1

√

H ̃( 22 −)b ̃ˆ−^12 (w ̃ˆM;q−wM;q). Then it holds that:

E(vˆ ̃vˆ ̃

′

)=E



(n−p)

Hˆ ̃(−)

22

(n−1)H ̃

(−)

22



E(cc′)=(n−p)

2



(n−p− 1

2

)



(n−p

2

) (Var(c)+E(c)E(c)′)

To evaluate the covariance matrix of the random vectorcwe apply the following formula
of conditional variance:

Var(c)=E(Var(c|(n−1)−^1 ˆ ̃b))+Var(E(c|(n−1)−^1 bˆ ̃)) (13.21)

From equation (26.20) it follows thatVar(c|(n−1)−^1 bˆ ̃)=Iand, respectively,

E(Var(c|(n−1)−^1 bˆ ̃))=I (13.22)

Let consider the second term in equation (13.21). It follows from the proof of part (a)
that

E(c|(n−1)−^1 ˆ ̃b)=

√

n− 1

√

H ̃( 22 −)b ̃−^12 T(W ̃M;q−wM;q)

whereTis a lower triangle random matrix.
Let us denoteW=(w ̃M;q−wM;q)(w ̃M;q−wM;q)′=(wij)i,j=1,...,q. Then

Var(E(c|(n−1)−^1 bˆ ̃))=H ̃

(−)

22 ̃b

−^12 E(TWT′)b ̃−^12 ′

−H ̃( 22 −) ̃b−

(^12)
E(T)WE(T′)b ̃−
12 ′
=H ̃( 22 −)b ̃−
(^12)
(G−F) ̃b−
12 ′
whereG=(gij)i,j=1,...,qandF=(fij)i,j=1,...,qwith
gij=E(
∑i
l= 1
∑j
k= 1
tilwlktjk)=wijE(tiitjj)







2 wii

(n−p+q−i+ 1
2 +^1
)

(n−p+q−i+ 1
2
),ifi=j
2 wij

(n−p+q−i+ 1 + 1
2
)

(n−p+q−i+ 1
2
) 
(n−p+q−j+ 1 + 1
2
)

(n−p+q−j+ 1
2
) otherwise
.