Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 243

space whose (theoretical) dimension is O(n ), where is typically less than

1=2, A 22 is usually invertible.
We now show that O n equals I n defined in Equation (9.10). With the
approximation space n in this section, we have

X

I n = n 1

[ ` 1 (X i ; b n ) ` 2 (X i ; b n )( b n )] 2 ;

i=1

where b n is the minimizer of

X

k` 1 (X i ; b n ) ` 2 (X i ; b n )(h)k 2

n (h; b n ) = n 1

i=1

over h 2 n for h = (h 1 ;h 2 ; ;h d ) 0 . Write h j = b 0 n c j for j = 1; 2; ;d.
Let Y ij = ` 1;j (X i ; b n ), the j-th component of Yi i = ` 1 (X i ; b n ), and Z i =
` 2 (X i ; b n )(b n ). This minimization problem becomes a least-squares problem
of nding fc n1 ;:::;c nd g with c nj 2 R q n that minimizes

X d X

(Y ij Z 0 i c nj ) 2 :

n 1

i=1 j=1

By standard least-squares calculation,

X ! X !

Z i Z 0 i

c nj = Z i Y ij ; 1 j d:

i=1 i=1

Hence, with b n = (b 0 n c n1 ;:::; b 0 n c nd ) 0 and by Equation (9.4), we have

h ` 1 (X i ; b n ) ` 2 (X i ; b n )( b n ) i 2

X

I n = n 1

i=1

2
4 Y i

3
5

X ! X !

2

X

= n 1 Y i Z 0 i Z i Z 0 i

Z i

i=1 i=1 i=1

X

Y i A 12 A 22 Z i 2

= n 1

i=1

= A 11 2A 12 A 22 A 21 + A 12 A 22 A 22 A 22 A 21
= A 11 A 12 A 22 A 21 = O n :

We summarize the above calculation in the following proposition.